New Approaches to Understand Constraint Effects on the ...

New Approaches to Understand Constraint Effects on the

Onset of Upper Shelf Temperature in a Reactor

Pressure Vessel Steel

A thesis submitted to the University of Manchester for the degree of Master of

Philosophy in the Faculty of Engineering and Physical Sciences

2013

Tony K Pramanik

Supervisor: Professor Andrew Sherry

Faculty of Engineering and Physical Science

School of Materials

New Approaches to Understand Constraint Effects on the Onset of Upper Shelf Temperature in a Reactor Pressure Vessel Steel

Table of Contents

Abstract .......................................................................................................................... 4

Declaration ..................................................................................................................... 5

Copyright Statement ..................................................................................................... 6

Acknowledgments......................................................................................................... 7

1.0 Introduction and Aims of Research ................................................................... 8

2.0 RPV Microstructure and Fracture mechanisms .............................................. 10

2.1 RPV Microstructure .......................................................................................... 10

2.2 Fracture Mechanisms ....................................................................................... 14

2.2.1 Lower shelf and lower transition behaviour- Cleavage Fracture ................ 15

2.2.2 Upper transition and upper shelf behaviour- Ductile tearing ...................... 16

2.2.3 The interaction between the mechanisms ................................................. 17

2.3 Knowledge Gaps .............................................................................................. 18

3.0 Fracture Mechanics ........................................................................................... 19

3.1 Background to Fracture Mechanics .................................................................. 19

3.1.1 Linear Elastic Fracture Mechanics ............................................................. 22

3.1.2 Elastic Plastic Fracture Mechanics ............................................................ 24

3.2 Constraint effects ............................................................................................. 27

3.2.1 T- Stress .................................................................................................... 27

3.2.2 Q-Parameter .............................................................................................. 28

3.2.3 Anderson and Dodds approach ................................................................. 29

3.3 Structural Integrity Assessment ........................................................................ 30

3.3.1 Master Curve Methodology ........................................................................ 30

3.3.2 Relationship between T0 and T-stress ....................................................... 34

3.3.3 J-R Curve Methodology ............................................................................. 36

3.3.4 Onset of Upper Shelf Temperature ............................................................ 37

3.3.5 Effect of constraint ..................................................................................... 38

3.4 Knowledge Gaps .............................................................................................. 39

4.0 Experimental Method ........................................................................................ 40


4.1 Master Curve .................................................................................................... 40

4.2 Materials Characterisation................................................................................ 40

4.2.1 Metallography ............................................................................................ 40

4.2.2 Transmission Electron Microscopy (TEM) ................................................. 40

4.3 Finite Element Analysis (FEA) .......................................................................... 41

4.3.1 Preliminary investigation of crack interaction for double crack specimens 41

4.3.2 Sensitivity analysis of testing specimens with different pre-crack lengths . 44

5.0 Results and Discussion .................................................................................... 48

5.1 Master Curve analysis ...................................................................................... 48

5.1.1 Analysis at T= -154oC ................................................................................ 50

5.1.2 Analysis at T= -110oC ................................................................................ 52

5.1.3 Analysis at T= -91oC .................................................................................. 52

5.1.4 Analysis at T= -60oC .................................................................................. 54

5.1.5 Analysis at T= -40oC .................................................................................. 56

5.1.6 Analysis at T= -20oC .................................................................................. 57

5.1.7 Analysis at T= -10oC .................................................................................. 59

5.1.8 Analysis at T= 0oC ..................................................................................... 60

5.1.9 Analysis at T= +20oC ................................................................................. 61

5.1.10 Interim Conclusions ................................................................................... 63

5.2 Materials Characterisation................................................................................ 64

5.2.1 Optical Microscopy .................................................................................... 64

5.2.2 Scanning Electron Microscopy .................................................................. 66

5.2.3 Transmission Electron Microscopy ............................................................ 67

5.2.4 Interim Conclusions ................................................................................... 67

5.3 Finite Element Analysis .................................................................................... 68

5.3.1 Preliminary investigation of crack interaction for double crack specimens 68

5.3.2 Sensitivity analysis of testing specimens with different pre-crack lengths . 80

5.3.3 Interim Conclusions ................................................................................. 108

6.0 Summary and Conclusions ............................................................................ 110

7.0 References ....................................................................................................... 112


Abstract The Reactor Pressure Vessel (RPV) is a critical pressure boundary component

in a nuclear power facility and is made of ferritic steel, which exhibits a change in its

failure mechanism of from brittle cleavage to ductile fracture with increasing

temperature. The RPV is designed to operate at temperatures where the material is

ductile under normal operating conditions and this is achieved by ensuring that when

stressed, the RPV temperature is above the Onset of Upper Shelf Temperature

(OUST). The OUST is defined as that temperature where the steel has a combined 5%

probability of cleavage and a 50% probability of ductile fracture, measured using high

constraint fracture toughness specimens. However, the OUST will reduce for defects

that have a lower level of crack-tip constraint, e.g. shallow cracks that exhibit reduced

levels of stress triaxiality. It is therefore important to develop new understanding of

constraint effects on the OUST for RPV steels.

This thesis is aimed at developing a new methodology with which to assess the

combined influence of constraint, microstructure and fracture mechanism on the

fracture toughness of RPV steels at temperatures near the OUST. A Master Curve

statistical analysis, and associated metallography, of the so-called Euro RPV material

was performed. Subsequently, a stress contour-based approach for studying constraint

effects on fracture was assessed and used within 3D design analyses of deep and

shallow-cracked fracture mechanics specimens loaded in four-point bending: (a) a

single specimen containing two offset cracks, (b) two single crack specimens loaded in

series (i.e. one on top of the other), and (c) two single crack specimens loaded in

parallel (i.e. side by side). The stress contour-based approach was shown to be a valid

approach for assessing constraint effects. The single specimen containing offset

cracks was shown to be the most favourable experimental approach for the following

reasons. Two specimens loaded in series limit to the maximum applied displacement

and exhibit frictional effects at the supporting rollers. Two specimens loaded in parallel

with slightly different crack lengths will have different compliances. With appropriate

design, a single specimen containing two offset cracks can be used to study the

constraint effects on OUST and the interaction between cleavage and ductile fracture

mechanisms. This experimental approach, combined with an analysis of the associated

crack-tip stress fields can provide a new approach for studying the influence of

constraint on the OUST in RPV steels.


Declaration

I, Tony K Pramanik, state that no portion of the work referred to in the thesis has been

submitted in support of an application for another degree or qualification of this or any

other university or other institute of learning.


Copyright Statement

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owns certain copyright or related rights in it (the “Copyright”) and s/he has given The

University of Manchester certain rights to use such Copyright, including for

administrative purposes.

ii) Copies of this thesis, either in full or in extracts and whether in hard or electronic

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Acknowledgments

The author wishes to acknowledge a number of people for their help and support

throughout this work:

Professor Andrew Sherry of the University of Manchester, for all of the help,

guidance, support and invaluable technical discussion.

Sincere thanks go to Dr Dan Cogswell and Dr Rob McCluskey at Rolls-Royce,

for helpful guidance during the course of the work.

Michael Faulkner for his expertise of SEM and TEM microscopy, and his

technical advice for which I am very grateful.

Thanks also go to Rolls-Royce and to the Engineering and Physical Sciences

Research Council for the provision of funding for this research.

On a personal note, I would like to thank my Mum, Dad and fiancée Josephine

for all their love and encouragement. Thank you


1.0 Introduction and Aims of Research

The Reactor Pressure Vessel (RPV) is the most critical pressure boundary

component in a nuclear power facility. It is the component that contains the nuclear fuel,

and so failure of this component cannot be tolerated. It is defined as an Incredibility of

Failure (IOF) component by the nuclear industry [2], for which safety cases must

demonstrate that failure of the component is an ‘incredible’ event with a frequency of

less than 10-7 per year of reactor operation.

Reactor Pressure Vessels are made of ferritic steel, which exhibits a change in

its mechanism of failure from brittle cleavage to ductile fracture with increasing

temperature. To ensure the safe continued operation of a nuclear power plant, failure

by brittle cleavage fracture of the RPV must not be allowed to occur. Cleavage fracture

occurs when operating temperatures are in the lower transition temperature regime, for

example when steel is embrittled by radiation, and is associated with low fracture

toughness. Cleavage fracture is catastrophic by nature, and has major consequences

on public safety and environmental contamination. Reactor Pressure Vessels are

designed to operate at temperatures where the material behaves in a ductile manner

under both normal operating and accident loading conditions. This is to ensure that any

cracks which may be present in the vessel, would extend in a ductile manner, and not

fail catastrophically by unstable brittle cleavage [3].

This is achieved by ensuring that when the RPV is significantly stressed, the

temperature is always in excess of the so-called Onset of Upper Shelf Temperature

(OUST). The RPV will become stressed by internal pressure during start-up and

shutdown of a reactor; hence it is critical that the applied pressure-temperature

relationship varies well below the boundary of brittle fracture for a given defect, as

illustrated in Figure 1-1. Significant stresses can also arise during operation if the RPV

needs to encompass a Pressurised Thermal Shock (PTS). Normal operating

temperatures must be in excess of the OUST, and this is normally defined with respect

to standard, high constraint fracture toughness data. However, the OUST will vary with

respect to defects with a reduced level of constraint, e.g. shallow cracks, so it is

important to understand the influence that crack tip constraint loss and ductile damage

has on fracture at this temperature. Constraint is a measure of stress triaxiality ahead of


the crack tip. The fracture toughness at which cleavage initiates decreases with the

increase of crack tip constraint, which occurs in larger specimens or that of specimens

that have a higher crack depth to width ratio.

Figure 1-1: Schematic showing the boundary of brittle fracture and operating conditions

This project is concerned with developing new understanding of the main factors

that influence the OUST in RPV steels.

The aim of the research is to develop a new methodology to assess the combined

influence of constraint, microstructure and fracture mechanism on the fracture

toughness behaviour of RPV steels at temperatures near the OUST.

Given this aim, the objectives of the research are:

1. To perform statistical analysis of transition toughness properties of the Euro

material, an RPV ferritic steel.

2. To undertake a basic metallographic examination of Euro material.

3. To test the validity of using a stress contour based approach to study the

influence of constraint on the OUST.

4. To design a fracture mechanics specimen to study the mechanism of fracture at

temperatures near the OUST where cleavage and ductile mechanisms interact.


2.0 RPV Microstructure and Fracture mechanisms

2.1 RPV Microstructure

Reactor Pressure Vessels are made substantially of ferritic steel and have typical

microstructures of bainite, tempered bainite, tempered martensite, ferrite and pearlite.

All ferritic steels have a body centred cubic crystal structure that displays ductile to

brittle transition temperature fracture toughness characteristics[4].

Typical RPV base alloys are A302B, A533 plates or A508 forgings, which are

quenched and tempered with primarily bainitic microstructures. Chemical compositions

are typically C(0.05–0.2%), Mn(0.7–1.6%), Mo(0.4–0.6%), Ni(0.2– 1.4%), Si(0.2–0.6%),

and Cr (0.05–0.5%)[5]

Depending on the relative weight percentage of carbon in the microstructure,

Figure 2-1 describes the iron-carbon system of alloys, and shows the phase boundaries

of the system. Steel with 0.2% carbon contains about 80% ferrite and 20% pearlite[6]

under equilibrium conditions.

Figure 2-1: Iron- carbon phase diagram


The primary control of the microstructure of low alloy steel is the cooling rate

from the austenite () phase. If the cooling rate is especially rapid, a finer grain structure

is formed, lending to better mechanical properties, such as higher fracture toughness,

strength and ductility. This can be shown in the Time Temperature Transformation

(TTT) diagram in Figure 2-2.

Figure 2-2: Time Temperature Transformation curve

A- austenite; B- bainite; M- martensite; P- pearlite

As the slow cool passes through the pearlite region of the TTT diagram, a

diffusional transformation occurs. As this is a gradual process, the microstructure has

time to organise and modify, which leads to moderately sized grains (~10-50μm) [6].

A faster cooling rate would bypass the pearlite 'nose', and the structure produced

is acicular ferrite that contains a smaller grain size (~ 1-10μm). This creates a bainitic

material, that improves all mechanical properties, such that both strength and fracture

toughness will increase, rather than compromising one for the other[6].


The fastest cooling would bypass the bainite nose, creating a martensitic

microstructure. This contains a finer grain size with the most improved strength

properties. Unfortunately, cooling at a rate of >50 °C/sec is largely unachievable for

reactor pressure vessels due to the thicknesses involved, typically 200mm thickness[7].

A widely studied RPV steel is the "Euro" material, which is termed 22NiMoCr37,

and is equivalent to AISI grade A508 Cl.2. It is a quenched and tempered vacuum

treated low alloy steel and likely to have been open die forged for RPV service (though

the material never went into operation). The quenching is necessary to suppress the

normal breakdown of austenite into ferrite and cementite, and to cause a partial

decomposition at such a low temperature, that martensite is produced.

This material is high in strength, typically used for pressure vessels, pressurisers

and steam generators in nuclear power plants[8].

The chemical composition of A508 Cl.2 is given in Table 2-1below[9].

C Si Mn P S Cr Mo Ni Cu

0.21 0.24 0.82 0.003 0.004 0.003 0.82 0.79 0.049

Table 2-1: Material chemical composition of A508 Cl.2 (wt.%)

Kim et al has shown the effects of alloying elements on mechanical and fracture

properties[10], and illustrated the typical RPV microstructure of A508 cl.3, Figure 2-3.


Figure 2-3: Optical micrograph of A508 Cl.3 steel, Nital etched [10].

Figure 2-3 displays a grain size of about 30 µm, and has an upper bainitic

microstructure. A508 Cl.3 is a low carbon ferritic steel, typically 0.2 wt% C. The

microstructure largely consists of ferrite (white) and carbon (dark).

A number of coarse carbides are formed in the bainitic ferrite matrix, seen in

Figure 2-4 below.

Figure 2-4: SEM micrograph of A508 Cl.3 showing carbide precipitation, Nital etched [10]


Work by Lee et al[11] has shown that the cleavage fracture toughness properties

measured in the transition temperature region can actually be interpreted using a

simple fracture model containing carbide size distribution. It was concluded that the

critical nearest neighbour distance between coarse carbides is an important

microstructural factor governing fracture toughness behaviour, since it satisfied a linear

relationship with the critical distance between a cleavage initiation site and a crack tip.

2.2 Fracture Mechanisms

The mechanism of fracture is similar in a variety of low alloy steels of various

compositions and is highly temperature dependant. In a broad context, four defined

regions can be defined across the transition temperature regime; these are the lower

shelf, lower transition region, upper transition region and the fully ductile upper shelf.

These regimes are illustrated in Figure 2-5[7] The ordinate describes a number of

fracture mechanics parameters which will be described in further sections, whilst the

abscissa defines the temperature relative to a reference temperature, T0. The

mechanisms of fracture associated with each region are described in more detail in the

following sections.

Figure 2-5: Schematic showing the regions of the Fracture Toughness transition regime


2.2.1 Lower shelf and lower transition behaviour- Cleavage Fracture

At lower shelf temperatures (region A, Figure 2-5), failure occurs by

transgranular cleavage. Fracture occurs by the initiation of cleavage from multiple sites

at the tip of the crack as the applied stress exceeds the cleavage stress[3]. These

microcracks inject into the surrounding matrix, and propagate through the remaining

ligament, illustrated in Figure 2-6a. It is a very fast fracture, requires a low amount of

energy, and hence will occur at low fracture toughness.

Within the lower transition temperature regime, fracture is initiated by the

cracking of hard second phase particles in the fracture process zone, ahead of the

crack-tip. The fracture toughness increases with temperature as more energy is

required to propagate the microcrack into the surrounding matrix material. Cleavage will

occur from a large number of possible initiators that are sampled within the volume of

the plastic zone, but ultimately the most potent initiates final failure. Cleavage failure will

originate from a single initiation site, illustrated in Figure 2-6b.

Cleavage fracture occurs due to the limited number of active slip systems,

attributed to the low temperature. Rapid propagation of a crack occurs under small

scale yielding conditions, and is most likely when plastic flow is limited. As a body-

centred cubic material, ferritic steel has a transgranular fracture path on {100}

crystallographic planes[12]. However, the crack propagation will change direction as it

crosses the grain boundary and enters the next grain in an attempt to grow in a {100}

plane that is close to perpendicular to the maximum principal stress[12].

Cleavage propagation is normally unstable, so the material near the tip of the

growing crack is subject to very high strain rates, which suppress plastic deformation.

To break bonds in the material microstructure, the concentration of local stress must be

sufficiently high to prevail over the cohesive strength of the material. Cleavage fracture

is initiated if the plastic zone encompasses an initiating microstructural defect such as a

carbide or pre existing microcrack[13]. If a uniform distribution of potential initiators is

assumed, the probability of fracture at a given temperature is governed by the volume

of the plastic zone and the applied stress. Therefore, a larger plastic zone under a

higher stress will sample a greater number of defects, so the greater the probability of


sampling a potent crack initiator. It is due to this that cleavage failure can be described

by the weakest link theory [14].

Understanding the fracture mechanisms of steel in the ductile - brittle transition

region requires analysis of crack initiation and propagation. Unstable cleavage cracks

can occur after blunting of the original fatigue precrack, or even after stable crack

growth, but in either circumstance, crack instability is believed to be triggered by

fracture of a brittle particle ahead of the crack. The scatter associated with KIC values in

the transition region is reflected in the size distribution of these so called "trigger"

particles[15].

2.2.2 Upper transition and upper shelf behaviour- Ductile tearing

In the upper transition region, increasing amounts of ductile damage and tearing

will precede cleavage fracture. As the crack grows by ductile tearing, more material is

sampled by the high stress region. Eventually the crack will sample a potent cleavage

initiator, and cleavage failure will prevail. Fracture in the upper transition regime exhibit

scatter, due to the distribution of second phase particles ahead of the crack tip. An

increase in fracture toughness is also observed, which is due to the greater amount of

energy required to create new surfaces, which will increase in proportion to the amount

of ductile tearing that precedes cleavage failure.

At upper shelf temperatures, failure will occur by ductile tearing. Figure 2-6c

schematically illustrates microvoid initiation, growth and coalescence at the tip of a pre-

existing crack. The voids grow as the crack blunts, and they eventually link with the

main crack.


2.2.3 The interaction between the mechanisms

In the upper transition regime, ductile tearing will precede cleavage fracture. The

cleavage fracture could be promoted by an increase in the local stress resulting from

the work hardening of the material at the tip of the crack[16]. At these temperatures and

at sufficiently high loads, gross section yielding of the specimen can cause the

relaxation of the triaxial stress state due to constraint loss, making it extremely difficult

to develop sufficient tensile stresses at the crack tip to initiate brittle fracture. It is before

the brittle fracture occurs, that the plastic strain at the crack tip becomes sufficiently

high to give rise to the initiation and propagation of ductile tearing. Milne and Curry [17]

proposed that ductile crack growth causes an increase in stress in the remaining

ligament. Due to this, a propagating crack in a ductile mode can lead to cleavage

instability after an amount of ductile crack growth.

The effect of increasing temperature, is that the critical value of cleavage fracture

toughness increases, whereas fracture toughness for ductile crack initiation and growth

in ferritic steel is substantially independent of temperature up to 100oC [18]. This can be

seen in Figure 2-7 were the probability distribution function of cleavage fracture

toughness values is displaced along the axis of fracture toughness with increasing

temperature; whereas the line representing the onset of ductile tearing occurs at the

same level of fracture toughness.

a: Lower Shelf b: Transition

Region

c: Upper Shelf

Figure 2-6: Schematic of fracture mechanisms in the transition regime


Figure 2-7: Variation in the probability of cleavage fracture with temperature [16]

2.3 Knowledge Gaps

As discussed in the previous section, the cleavage fracture mechanism is well

understood[19–21], and the mechanism of fracture in the ductile regime is also well

known[16, 21]. The knowledge gap arises from the limited understanding of the

interaction between the cleavage fracture and the ductile fracture mechanisms within

the onset of upper temperature regime. The material microstructure permits both

mechanisms to compete due to the involvement of second phase particles (e.g.

carbides) as potential initiating particles for either cleavage or ductile fracture. A

particular question that arises is:

1. What are the key microstructural features that influence failure in the onset of

upper shelf region of RPV steels?


3.0 Fracture Mechanics

The scope of this fracture mechanics section is as follows. A review on the

background to the subject is provided in section 3.1. This includes Linear Elastic

Fracture Mechanics (LEFM) in section 3.1.1 and Elastic Plastic Fracture Mechanics in

section 3.1.2.

Section 3.2 describes the effect that constraint has on the fracture toughness of a

specimen, and two parameters that govern constraint, the T stress and Q-parameter

are described in section 3.2.1 and 3.2.2 respectively.

Section 3.3 discusses the importance of accurate structural integrity assessment, and

reviews the use of the Master Curve methodology in section 3.3.1. The relationship

between the Master Curve transition temperature T0 and the constraint parameter T-

stress is highlighted in section 3.2.1. This leads onto J-R curve methodology which is

described in section 3.3.3, providing necessary background to understanding the

parameters that effect the Onset Of Upper Shelf Temperature (OUST), which will be

discussed in section 3.3.4. The effects of constraint on the OUST will be discussed in

section 3.3.5.

Finally knowledge gaps will be addressed in section 3.4.

3.1 Background to Fracture Mechanics

Early studies in fracture mechanics by Griffith[23] in the 1920s proposed that

cracks can only be formed or propagate if the process causes the total energy in the

system to decrease or remain the same.

If the crack tip of a defect is truly sharp, with a negligible radius of curvature, then

the concentration of stress experienced at the crack tip is infinitely large and an

infinitesimally small load would cause an infinite stress and hence failure. In response

to this, Griffith developed an energy based theory of fracture, rather than a purely stress

based approach.

Crack propagation is the result of excess potential energy that overcomes the

surface energy of the material, and for a small increase in crack area dA, the Griffith


energy balance equation can be expressed as:

(3-1)

where

γ= total energy

A=crack area

Π= potential energy from external forces and strain energy

Ws= energy to create new surfaces

Due to the equilibrium nature of fracture, dE/dA = 0 (as there is no change in the total

energy of the event) and as critical new surfaces are created after crack formation,

Griffith derived the equation for the stress of fracture σf, to cause brittle fracture from a

starting crack with an internal crack length, 2a:

(3-2)

E= Young's Modulus

γs= Surface energy

Griffith's research focussed on the fracture characteristics of glass, however

Irwin[24] adapted the Griffith energy equation to account of materials that are not ideally

brittle, providing a more accurate expression for the fracture strength of metal. By

including the energy of plastic work per unit area γp, the modified energy balance

equation becomes:

(3-3)

Where the effective surface energy is γeff = γs + γp for quasi-brittle elastic plastic

material.


Irwin built upon Griffith’s approach, which is more suitable for addressing real flaws[25].

The energy release rate, G is defined as the measure of energy that causes an

increment of crack extension, where:

(3-4)

As G can vary with crack size, Irwin showed that to predict the stability of a crack, G is

equal to the materials resistance to crack extension R, and for a stable crack,

(3-5)

and for an unstable crack:

(3-6)

Analysis of R curves and their shape varies significantly for brittle and ductile

fracture; due to plastic zone formation in a ductile body, the external force must

increase to maintain steady crack growth, giving a rising R curve. For a brittle material,

cleavage fracture is characterised by a falling R Curve because it is a very fast fracture,

exhibiting less resistance to extend the crack at the onset of fracture.


3.1.1 Linear Elastic Fracture Mechanics

Westergaard and Irwin developed expressions for stresses in the vicinity of an

infinitely sharp crack in a specimen subjected to an external force[17, 18]. If a material

exhibits isotropic linear elastic behaviour (i.e. its physical properties have the same

parameters in different directions) they predicted that there will be a fundamental stress

field proportional to 1/√r, where r is the distance from the crack-tip. This describes a

stress singularity as it is asymptotic as r approaches the crack tip, i.e. r 0. Figure 3-1

below shows the conventional axis system of the stress fields ahead of the crack tip.

Figure 3-1: Conventional polar coordinate axis system of the stress fields ahead of the crack tip

The tensile stress field σyy approaches infinity as r→0, despite the other terms

remaining finite or tend to a value of zero. This can be expressed as:

(3-7)

where KI is the stress intensity factor due to Mode I loading, however Figure 3-2 shows

loading can occur in three ways.


Figure 3-2: Crack loading modes of opening, shear (in plane) and shear (out of plane)

Westergaard calculated the stress fields ahead of a crack tip for all modes of loading in

a linear elastic, isotropic material, known as the Westergaard functions. The stress field

formula for the σyy direction is given below, where any angle of θ may be used [26].

(3-8)

The stress intensity factor, KI, can be calculated for a number of specimen types

such as Single Edge Notched Bend SEN(B), Double Edge Notch Tension DEN(T) or

compact tension C(T). The KI solution for a Compact-Tension C(T) specimen is given in

the expression below taken from Paris et al [28].

(3-9)

where

P = Load

W = Specimen width

B = Specimen thickness.

Mode I Mode II Mode III


3.1.2 Elastic Plastic Fracture Mechanics

Elastic Plastic Fracture Mechanics is applicable when non-linear material

deformation is contained to a small region at the crack tip, known as small scale

yielding conditions (SSY).

The Crack Tip Opening Displacement (CTOD) describes the degree of crack

blunting that occurs as the crack faces move apart due to plastic deformation[29]. The

work by Irwin states that the CTOD can be estimated by solving the displacement that

occurs due to crack tip plasticity, from the viewpoint that the blunted tip can be

extrapolated into the plastic zone. The Irwin plastic zone correction can be used to

calculate the effective crack length under plane stress conditions [30]:

(3-10)

where

ry = Effective crack length within the plastic zone

σYS = Yield stress of the material

Definitions of CTOD can be illustrated in Figure 3-3. If the crack blunts in a semi-circle,

these two definitions have the same value.

The opening displacement of the original crack tip

The displacement at the

intersection of a 90o

vertex with the crack

flanks

Figure 3-3: Definition of Crack Tip Opening Displacement


Another crack driving force parameter is the J contour integral that Rice

developed by representing elastic-plastic deformation as non-linear elastic, and

expanding the field of fracture mechanics beyond the limits of LEFM[3]. The loading

behaviour is the same for both material types; however the unloading effect differs

between the two. Non-linear elastic materials follow the same stress-strain law during

loading and unloading behaviour, but the elastic plastic material exhibits irreversible

plasticity, and follows a linear path related to Young's modulus [31].

Expressed as an energy parameter, the rate of change of potential energy with respect

to a crack length a is given by:

(3-11)

For linear elastic materials J = Ԍ, and its relationship with K is:

(3-12)

The J integral can be written as a path independent line integral. If a line, Г follows an

anticlockwise movement around the crack tip such as in Figure 3-4, then the J integral

can be expressed as:

(3-13)

where

w= strain energy

n= Components of the traction vector

ui= displacement vector components

ds= length increment along the contour Г.


Figure 3-4: Path independent Line integral around crack tip

Expressed as a stress intensity parameter, the papers by Hutchinson[32] and

Rice and Rosengren[33] show that crack tip conditions in a non-linear elastic material

can be uniquely characterised by J. For a material that follows the Ramberg-Osgood

stress-strain law:

(3-14)

for uniaxial deformation, where

σ0= Reference stress (normally equal to yield stress)

Ɛ0= σ0/E

α= Constant (No units)

n= Strain hardening exponent of the material.

Where n=1 for a linear elastic material, and 5<n<10 for ferritic steels[34]. The

HRR field equation [35] demonstrates that stress and strain vary 1/r near the crack tip

for n=1. For a Ramberg-Osgood material, the stress fields close to the crack tip are

shown as:

(3-15)

and the strain fields:


(3-16)

where

In= Integration Constant dependant on n

ỡij and έij= Functions of θ and n (no units)

These are known as the HRR field solutions, named after Hutchinson[16,20], Rice and

Rosengren[33].

3.2 Constraint effects

Under SSY conditions, there is a high concentration of local stress triaxiality

ahead of the crack tip. For this circumstance, the fracture parameter J can accurately

describes the crack tip stress fields via the HRR field. Fracture toughness testing

standards[28, 29] specify size requirements for test specimens to ensure fracture

occurs under SSY conditions, i.e. where J is a valid crack tip parameter.

Under low constraint conditions, the development of plasticity leads to reduced

stress triaxiality conditions ahead of the crack than is predictd by J. This loss of crack

tip constraint occurs as yielding develops beyond SSY, and may be described as Large

Scale Yielding (LSY), i.e. where the plastic zone is a large proportion of the uncracked

ligament or extends towards the free surfaces of the specimen[39].

Several parameters that quantify constraint loss are reviewed in the next

following sections.

3.2.1 T- Stress

The T-Stress is an important feature of the asymptotic nature of the stress field

ahead of a crack tip under linear elastic conditions. Following the work of Williams[40],

the expansion of the elastic stress field close to the crack tip in mode I loading may be

written as:

(3-17)


Where fij(θ), Tij(θ) and Uij(θ) represent the angular variation of the field. The first term is

singular in r. The second term, the T-stress, remains finite near the tip. The remaining

terms vanish at the crack tip as r→0 [41].

The T- stress represents the stress acting parallel to the crack plane. T-stress

has a considerable influence on plastic zone shape and size that develops at the crack

tip[33, 34]. A negative value of T indicates a low constraint condition, whereas for high

constraint conditions, T is positive.

3.2.2 Q-Parameter

The Q-parameter, derived from the HRR stress field, have been used to characterise

the level of constraint present in cracked bodies[22], [44], [45]. The Q-parameter is a

dimensionless quantity, defined by the following relation:

(3-18)

where

(σij)HRR = HRR stress field (equation 15)

σij = Stress field ahead of crack tip

Q can also be evaluated as the difference of the SSY reference solution and the

actual stress field by[22]:

(3-19)

For θ = 0 and at the normalised distance r= 2J/σ0. Geometries with Q< 0 have a low

triaxial stress state ahead of the crack tip, loss of J-dominance and therefore low

constraint.

Geometries with Q ≥ 0 display high stress triaxiality, strong agreement with HRR stress

fields and are as a result, highly constrained [46]. Conversely, a negative value of Q

indicates a loss of constraint.


3.2.3 Anderson and Dodds approach

Single edge-notched bend (SENB) specimens containing shallow cracks are

commonly employed for fracture testing of ferritic materials in the transition region

where critical J (JC) values for shallow crack specimens are significantly larger than the

JC values for corresponding deep crack specimens at identical temperatures. The

increase in fracture toughness is due to the loss of constraint that occurs when the

plastic zone interacts with the tensile surface, causing a decrease in the stress triaxiality

at the crack tip. Anderson and Dodds [47] developed an approach to correlate the effect

a/W ratio has on crack tip stress contours, with elastic plastic fracture toughness (JC),

seen in Figure 3-5. By comparing the J values from SENB specimens against J values

for the SSY model, at equivalent crack tip opening mode stress, the increased JC for the

shallow crack can be predicted.

Figure 3-5: Schematic of the relationship between the J-integral values in SENB specimens for deep and

shallow cracks with those of SSY conditions, at equivalent crack tip opening mode stress[47].


3.3 Structural Integrity Assessment

There are a number of statistical approaches for characterising fracture

toughness properties in the transition regime. These include the Moskovic competing

risk analysis[48], local approach[49], toughness scaling [39] and the Master Curve [37],

[50]. This section describes the most widely used approach: the Master Curve

methodology.

3.3.1 Master Curve Methodology

The Master Curve method, developed by Wallin at VTT Manufacturing

Technology, is used to describe the statistical distribution of fracture toughness test

results. The standard ASTM 1921-10[37] is the widely accepted approach for

determination of the reference temperature T0. This is the test temperature at which the

median of the fracture toughness distribution equals 100MPa√m. It has been shown

that the fracture of steels is of a statistical nature, therefore the scatter in KJc results can

be modelled statistically, using a Weibull distribution [14].

The two parameter Weibull distribution gives the failure probability as:

(3-20)

where

KI = applied stress intensity

K0 = normalization factor (scale parameter)

m1= describes scatter variance (shape parameter)

If the magnitude of scatter is large, then m is small and vice versa. Experimentally, the

shape parameter has the range of 2-10 [14]. The three parameter Weibull distribution,

introduces a limiting value, Kmin, beneath which cleavage propagation is deemed to be

impossible, and is added into the three parameter failure probability as:

(3-21)


It can be seen from Figure 3-6 that experimental values for the test result scatter are

within the theoretical expectance values, and the shape parameter converges to 4. As

the data fall within the confidence limits, the data does not disprove this assumption.

Figure 3-6: Weibull shape parameter as a function of the number of tests.

Due to a larger specimen having greater probability of sampling a cleavage

initiator, different size specimen data must be thickness adjusted for the calculation of

the transition temperature, T0. Thickness is adjusted to the reference flaw length B0 =

25mm using the equation below:

(3-22)

B0 = reference thickness (25mm)

B = nominal Thickness.

KJC data must be ranked, and rank probabilities assigned. These are estimates of

cumulative probability based on order statistics. Three common rank probability

estimates are [51]:

(3-23a)


(3-23b)

(3-23c)

However Lipton and Sheth [51] show that equation 23c provides the best estimate of

median rank probability when compared with a binomial theory estimate, as shown in

Figure 3-7 below.

Figure 3-7: Comparing median rank function with binomial theory estimate (Circles)

The Weibull fitting parameter, K0 is located at the 63.2% cumulative

failure probability level. This is derived from Equation 22, when KJC = K0, the Probability

of failure, Pf = 0.632. The determination of K0, when all data is valid is given by [37]:

(3-24a)

where

N = number of specimens tested

Kmin = 20MPa√m (in accordance to ASTM 1921-10)


or when data is violated (either by exceeding 0.05(W-ao) or 1mm stable crack growth),

then K0 can be expressed as:

(3-24b)

where

r = number of valid data only

N = All data, valid and invalid.

K0 can be converted for finding the 50% cumulative failure probability level of the data

population, called K Jc(med), by:

(3-25)

The reference temperature, T0 can now be calculated from either the scale

parameter K0 or KJc(med) by

(3-26a)

or

(3-26b)

An example of the Master Curve methodology is shown inFigure 3-8 below. The

data are taken from Rathbun et al [52] for a A533B Class 1 steel plate. The T0

reference temperature is highlighted, and the median fracture toughness curve is

presented as the solid line. The 95% and 5% probability bounds are presented on the

graph as the upper and lower dotted lines respectively.


Figure 3-8: Example of the Master Curve methodology, with the T0 reference temperature shown[52], all data

is thickness adjusted.

3.3.2 Relationship between T0 and T-stress

The crack tip stresses and strains that cause plastic flow and fracture in

components differs to that in test specimens. This is due to the constraint effect, which

is generally lower for shallow defects in RPV components than for deep cracks in test

specimens. This leads to fracture toughness values obtained via specimen data to be

conservative in comparison to the fracture toughness qualities of cracks in the RPV[53].

For deeply notched specimens, and a given level of loading, the T stress is positive and

Q is close to zero. Both parameters become negative as the crack length to thickness

ratio a/W ratio reduces. The reduction is due to an associated reduction in the

hydrostatic stress local to the crack tip.

Lidbury et al [53] noted that values of T0 for C(T) specimens tend to be 10 to

15oC higher than those for deeply cracked specimens of the same material. This is due

to the in-plane constraint of C(T) specimens being greater than that of an SE(B)

specimen with the same a/W ratio. By quantifying the influence of in-plane constraint on

T0, the relationship between T0 and T-stress can be seen in Figure 3-9. The data had

been collated from cleavage fracture mechanics tests, and in the case of C(T), SE(B)


and SE(T) geometries Ω= T stress/ σy is calculated at limit load using T stress solutions

of Sherry et al[54].

Figure 3-9: Correlation between measurements of ∆ T0 = T0, specimen - T0, CT

compared with ∆ T0 = T 0, Ω - T0, CT , as a function of Ω.


3.3.3 J-R Curve Methodology

With structural integrity assessments of real or postulated defects, it is important

to understand the fracture resistance of the material[55]. In the ductile regime, fracture

resistance is characterised by the J-R curve that relates fracture toughness Jmat to the

amount of ductile crack extension. Ductile tearing is believed to have initiated when the

crack driving force is equal to or greater than the value of Jmat at 0.2mm of stable crack

growth including blunting. The rising J-R curve, seen in Figure 3-10 is associated with

the ductile fracture mechanism, i.e. the growth and coalescence of microvoids. In the

initial stages of deformation, there is a small amount of apparent crack growth due to

blunting. As J increases, material at the crack tip fails locally, and the crack can re-

sharpen, and propagate in a stable manner. The J-R curve depicts this as the value of

applied J for crack extension rises as a function of crack growth. Of course, at the onset

of upper shelf temperature regime discussed in the next section, instability of the

growing crack can be encountered by the intervention of brittle cleavage.

Figure 3-10: An example J-R curve for a ductile material [55]


3.3.4 Onset of Upper Shelf Temperature

As noted previously, RPV's are designed to operate at temperatures where the

material behaves in a ductile manner under accidental loading. This ensures that any

crack which may be present in the vessel would extend in a ductile manner, and not fail

catastrophically by unstable brittle cleavage.

This is achieved by ensuring that when the vessel is under normal operating

conditions, its temperature is always in excess of the OUST, which is defined as a

temperature where the 5% probability of cleavage curve intersects the 0.2mm ductile

tearing curve[56]. This region can be seen from Figure 3-11, which shows a Master

Curve analysis of cleavage fracture toughness in the transition regime of A508 Cl.2

material and a normal distribution of ductile initiation toughness.

Figure 3-11: Variation in cleavage fracture toughness of A508 Cl.2 with increasing temperature.

Data analysis from [57].


3.3.5 Effect of constraint

The probability of cleavage fracture at a given temperature is related to the level

of constraint, and therefore varies with specimen size and geometry. The fracture

toughness at which cleavage initiates decreases with the increase of crack tip

constraint in larger specimens, or that of specimens that have a higher crack length to

width (a/W) ratio. The OUST will also be affected by geometry factors such as a/W

ratios. Figure 3-12illustrates the effect that deep and shallow crack lengths have on the

toughness- temperature relationships typical of a mild structural steel.

Figure 3-12: Schematic Toughness-transition curves for shallow and deep notch specimens [47]

Due to the lower crack tip constraint, the shallow notch will exhibit higher

apparent fracture toughness than the deep notch at a given temperature. The constraint

loss in shallow notch specimens can also cause a shift in fracture mode, as the

geometry permits the fracture process zone to interact with free surfaces. The deep

notch specimen however will have crack tip stresses and strains that will increase in

proportion, in accordance to the highly constrained HRR stress field derived by

Hutchinson[32], Rice and Rosengren[30, 32]. The loss of crack tip constraint on the

shallow notch reduces the intensity of the maximum principal stress field ahead of the


crack tip. This will increase the fracture toughness of both the cleavage initiation and

ductile initiation distributions, and cause a shift in the OUST, to higher fracture

toughness and towards lower temperatures, illustrated in Figure 3-13.

Figure 3-13: Schematic showing the shift in OUST for a low constraint geometry

3.4 Knowledge Gaps

As discussed in the previous section, the effect of constraint on brittle fracture

and upon ductile tearing is well understood [39], [44], [45], [47]. However, there is not a

full understanding of how constraint influences the onset of upper shelf behaviour as

illustrated in Figure 3-13, nor is there a combined experimental and modelling approach

that can be used to assess this effect.

0

50

100

150

200

250

300

350

400

-160 -110 -60 -10 40

Size

Co

rrec

ted

To

ugh

nes

s, K

JC (

MP

a√m

)

Temperature (oC)

Shift in OUST for low constraint geometry OUST for high

constraint geometry


4.0 Experimental Method

The experimental method is outlined in the following sections. The use of the

Master Curve methodology is described in Section 4.1. The methods used for analysing

the microstructure of euro-material are outlined in section 4.2 and numerical modelling

analysis is described in section 4.3.

4.1 Master Curve

The Master Curve methodology has been applied to over 750 fracture tests of

A508 Cl.2 [50]. The data are the result of a round robin collaboration of numerous

testing houses. Tests were performed on C(T) specimens in accordance with standard

E1921-10[37], with widths of 25, 50, 100 and 200mm. The a/W ratio was 0.6 for all

specimens.

4.2 Materials Characterisation

4.2.1 Metallography

Metallographic samples of A508 Cl.2 RPV steel were extracted from a section of

the Euro material forging. One side of the sample was cut by water jet machining, and

three samples were cut from the centre region of this 'slice' to ensure the microstructure

in each sample underwent the same treatment. These three samples where then

mounted in different planes for analysis- the S-T plane, L-T plane and L-S plane.

The samples where ground and polished to a µm finish then OPS polished

and etched with 2% Nital for 3-5 seconds. The microstructures were analysed using

optical microscopy and Scanning Electron Microscopy (SEM).

4.2.2 Transmission Electron Microscopy (TEM)

Samples were prepared using the extraction replica technique [58–60]. Samples

where first polished and etched, as describes in the previous section. The surface of the

sample was then spluttered with a thin carbon coating approximately 150nm in

thickness. This surface was then scored with a scalpel blade into 2mm squares and left

in 2% Nital etchant overnight. The etchant dissolved the surface of the metal, allowing


the carbon extraction replicas to float away from the sample. Using a pipette that had a

broad tube, the extraction replica was taken out of the etchant and cleaned in deionised

water twice. Needle nosed tweezers were then used to hold the copper grids to catch

the floating extraction replicas from the water. The extraction replicas were then dried

on filter paper.

4.3 Finite Element Analysis (FEA)

This section describes the Finite Element Analyses (FEA) undertaken to help

assess the viability of the three test approaches by modelling the testing of specimens

at temperatures close to the upper shelf. The preliminary analyses, described in Section

4.3.1 investigate whether two offset cracks in a single specimen would interact, when

loaded under four-point bending. The data generated from the double crack models

were compared to single crack models to assess whether cracks behaved

independently when located in close proximity to each other. Crack lengths assessed

were a/W = 0.5 (high constraint) and 0.1 (low constraint).

Section 4.2.2 describes the more detailed 3D FEA to assess the three test

approaches, to ensure that two cracks can be tested under the same loading conditions

to failure. To assess experimental issues, analyses were performed of specimens with

different pre crack lengths, as pre-cracking specimens accurately can present

challenges. The crack lengths analysed here are a/W = 0.45 and 0.55.

4.3.1 Preliminary investigation of crack interaction for double crack

specimens

Preliminary FEA analyses investigated crack interaction on the offset cracked

specimens, loaded under four-point bending. The data generated from these "double

crack" models were compared to single crack models to assess whether the two cracks

behaved independently of each other when located in reasonable proximity to each

other.

The geometry of all models followed the schematic diagram shown below in

Figure 4-1. Distances between pins were kept constant for both the single crack

specimen geometry and double crack. The crack length, a, was 12.5 mm or 2.5mm to


provide a/W ratio's of 0.5 or 0.1 respectively. Double cracked specimens had cracks

spaced at 28.3 mm apart for even spacing between crack mouths and the neighboring

pins. The specimens were assumed to be square sectioned with width W = B = 25mm.

The element types used were CPE4R. These are 4-noded, bilinear plane strain

elements using reduced integration for hourglass control.

The crack tip was set as a collapsed element, with duplicate nodes. The crack tip

had mesh refinement of a very small length of 0.003 mm. This was to ensure sufficient

resolution is present to display crack tip stress fields at low values of J. The offset crack

geometries for the specimens with a/W = 0.5 and 0.1 can be seen in Figure 4-2 and

Figure 4-3, respectively. Symmetry conditions were applied along the central axis of the

specimen and the support pins where given frictionless tangential behaviour.

Material properties were the same used by Anderson and Dodds[47] so the

results could be validated against results in the literature. The material properties

followed the Ramberg-Osgood hardening curve[35] defined by Equation 3.14 with

parameters provided in Table 4-1.

Figure 4-1: Standard Geometry of all specimens modelled (dimensions in mm)

S1 =190

S2 = 85

S2 = 85

S1 =190

Crack length, a

28.3

25

25


Figure 4-2: Mesh design for a/W = 0.5 SEN(4PB) model. Offset crack from the symmetry plane

Figure 4-3: Mesh design for a/W = 0.1 SEN(4PB) model. Offset crack from the symmetry plane

Yield stress, σy

(MPa)

Modulus, E

(MPa)

α

n

414 200,000 1 10

Table 4-1: Material Data used by Anderson and Dodds [47]


4.3.2 Sensitivity analysis of testing specimens with different pre-crack

lengths

This investigation built on the previous 2D FEA to focus on the feasibility of three

possible experimental designs. The specimens followed the geometry shown in Figure

4-4. The experimental designs are:

1. A single specimen with two cracks, offset from each other, under a constant

bending moment in four point bending, Figure 4-5.

2. Two specimens under load in series, where each specimen has a single crack.

The top specimen has the crack facing upwards, and the bottom specimen has

the crack facing downwards. This will cause the opening of both cracks during

test, Figure 4-6.

3. Two specimens under load in parallel, where each specimen has a single crack.

This can be seen in Figure 4-7.

Pre cracking specimens accurately to the desired crack length can present challenges;

hence this analysis investigated the sensitivity of testing specimens of slightly different

pre crack lengths, of a/W = 0.45 and 0.55.

Figure 4-4: Standard Geometry of the specimens modelled (dimensions in mm)

S1 =290

S2 = 100

S2 = 100

S1 =290

Crack length, a

55

25

25


Figure 4-5: Mesh design for a/W = 0.45 and 0.55 (on the left and right respectively) SEN(4PB) Model.

Evenly spaced offset cracks

Figure 4-6 Mesh design for a/W = 0.45 and 0.55 (on the top and bottom respectively) SEN(4PB) Model.

Single cracks specimens in series, tested one on top of the other.


Figure 4-7: Mesh design for a/W = 0.55 and 0.45 (in front and behind respectively) SEN(4PB) Model.

Single cracks specimens in parallel, tested next to each other .

The models created were 3D and developed using ABAQUS software[1].

Meshing of the four models followed a very similar procedure for consistency. The

element types used were C3D8R. These are 8 noded, linear brick elements using

reduced integration for hourglass control.

The crack tip was located on seamed edges in Figure 4-5 and on the symmetry

plane for Figure 4-6 and Figure 4-7. Assigning a seamed edge in ABAQUS will replace

a single node with two, allowing the two nodes to separate during loading simulating

either sides of the crack flank. This allows a crack length to be modelled without

symmetry. Figure 4-5 and Figure 4-6 had symmetry in the z-direction, allowing Figure

4-6 to be a quarter model representing the full specimen. Figure 4-5 and Figure 4-7 are

half models, and hence required further restriction in the x and z direction respectively.

This was achieved by restricting a single node opposite the crack face. The crack tip

was modelled as duplicate nodes collapsed as a single node, which allows the

collapsed nodes to displace during loading, which is effective to simulating crack tip

blunting.

Loading and support pins were modeled as discrete rigid surfaces, and were

given frictionless tangential behavior, and 'hard' contact as normal behavior. The single


specimen model (Figure 4-5) and the specimens in parallel (Figure 4-7) had fixed S1

pins, and the S2 pins were displaced by 10mm in the vertical {U2} direction, but

restricted in the horizontal {U1} direction and rotation {UR3}. The specimens in series

(Figure 4-6) had fixed S1 pins, and the S2 pins were allowed freedom in the vertical {U2}

direction, but restricted in the horizontal {U1} direction and rotation {UR3}. The top pins

in this model were displaced 10mm in the vertical {U2} direction, but restricted in the

horizontal {U1} direction and rotation {UR3}.

The material properties referenced in the finite element analysis were the same

properties used by Anderson and Dodds [47], as the material properties are similar to

A508 Cl. 2, and the results obtained could be validated against the results from the

literature. The material followed the Ramberg-Osgood stress-strain law defined by

Equation 3.14 [35]. The material properties used are summarised in Table 4-1.

The stress-strain behaviour of the material can be seen Figure 4-8 below.

Figure 4-8: Stress-Strain behaviour for the Ramberg-Osgood parameters in [35].

0

200

400

600

800

1000

1200

1400

1600

0 50 100 150 200 250 300

Stre

ss (

Mp

a)

Plastic Strain


5.0 Results and Discussion

The results section will be presented as follows:

The Master Curve analyses are presented in section 5.1. Optical microscopy,

SEM and TEM micrographs of the Euro material are presented in sections 5.2.1, 5.2.2

and 5.2.3 respectively. The FEA results are presented in section 5.3. This includes the

results of the preliminary 2D analyses of crack interaction for double crack specimens

compared with results for the single crack specimen model, in section 5.3.1. The

sensitivity analyses follow, in section 5.3.2 with regard to testing specimens with

different pre crack lengths in the three test configurations, under four point bending.

5.1 Master Curve analysis

The fracture toughness properties of the A508 Cl.2 Euro material are presented

in Figure 5-1 below. This shows over 750 data points generated from specimens of

thickness 12.5, 25, 50 and 100mm. This clearly shows the scatter, especially in the

upper shelf region. The specimen sizes of 50 and 100mm thickness exhibit values of

fracture toughness that tend towards the lower end of the distribution. This is due to

high constraint, and the larger fracture process zone in these specimens compared with

the smaller test samples. Larger specimens are more likely to sample material defects

near a crack tip, such as carbides due to the longer crack front, and are therefore more

likely to sample a cleavage initiator at a given level of crack driving force.


Figure 5-1: "Raw" fracture toughness data

As the dataset included ductile tearing preceding brittle fracture (measured

optically), the fracture toughness is plotted as a function of pre-cleavage tearing. Each

data point represents a test that failed in cleavage after some tearing, Figure 5-2. The

data all follow the same tearing resistance curve, showing no discernable size effect,

except that the larger specimens tend to fail at lower values of KJ, i.e. after less tearing,

than the smaller specimens. All data lie on a single J R-curve.


Figure 5-2: Multi specimen data showing comparison of ductile tearing and KJ.

5.1.1 Analysis at T= -154oC

The single temperature data are presented in a failure probability diagram that

displays a linear representation of the Master Curve. This offers a better visualisation of

the data with respect to the Master Curve, with the Weibull modulus set to equal 4. The

following subsections present a repeated Master Curve analysis of the Euro material

that has previously been performed by Wallin[50]. All single temperature analyses are

comparable to his analyses.

At this temperature, material behaviour corresponds to the lower transition

region. It is clear that the data follows the Master Curve prediction reasonably well. As

seen in Figure 5-3, all three cases look strikingly similar, and the size effect is not

clearly visible.

In all cases the gradient predicted by the Master Curve is slightly too steep

compared with the data, with approximately two-thirds of the data lying above the

Master Curve line at intermediate values of KJc. At high values of KJc, the data tend to

fall below the line.

0

200

400

600

800

1000

1200

1400

0 5 10 15 20 25

KJ M

Pa√

m

∆a (mm)

B= 12.5 mm

B= 25 mm

B= 50 mm

B= 100 mm


Figure 5-3: Failure Probability Diagram for 12.5, 25 and 50mm thick specimens at T= -154oC.



One specimen thickness of 12.5 mm was tested at -110oC. The material is

operating within the lower transition region. There are 55 specimens in this dataset, and

they follow the Master Curve prediction very well, seen in Figure 5-4. As in all analysis a

fixed Kmin was used of 20MPa√m. In this case the data are scattered around the Master

Curve line.

Figure 5-4: Failure Probability Diagram for 12.5mm thick specimens at T= -110oC


All the specimen sizes were tested at -91oC, operating in the lower shelf

transition regime. The results are shown in Figure 5-5.

For the 50mm thick specimens, the fracture toughness behaviour shows some

discrepancy from the Master Curve prediction, which could be due to the difficulty of

recognising initiation sites correctly on large fracture surfaces[50]. Slight discrepancies

in the 100mm thick specimen data could be due to a statistical sampling effect (as 15

specimens is regarded as a small sample for Weibull statistics[50]).

Essentially, the applied Master Curve prediction produces a reasonable

description of the data. The fixed Master Curve value of Kmin = 20 MPa√m was applied

to all data.


Figure 5-5: Failure Probability Diagram for 12.5, 25, 50, and 100mm specimens at T= -91oC


Three specimen thicknesses where tested at -60oC. All follow the Master Curve

distribution reasonably well. For the 12.5mm specimens, two data points violated the

KJc limit, (the maximum KJc capacity of a specimen) as set out in the standard

ASTM1921-10 [37]. This has the equation:

(27)

where

E=Young's modulus,

bo= Specimen Thickness

M= Non dimensional measure of constraint, ( = 30 in ASTM 1921-10[37])

The KJc limit can be seen for 12.5mm specimens at T= -60oC, and above this

limit the fracture toughness data lie to the right of the Master Curve, possibly due to a

loss of constraint or an increase in pre-cleavage tearing. The limit line sets the limit at

which the fracture parameter J provides a good description of the stress field close to

the crack tip (equation 15). Above the limit, J begins to be less valid as a characteristic

parameter with the actual stress field deviating from the HRR field by more than 10%[3].


The expected size effect is not in accordance with weakest link behaviour, as the 50mm

specimens show higher upper bound toughness than the smaller specimens. This is

also not explainable via constraint methodology. Wallin suggests this is possible due to

a macroscopic material variability [50], which is plausible due to the size of an RPV

forging.


Figure 5-6: Failure Probability Diagram for 12.5, 25, and 50mm specimens at T= -60oC


Toughness data have been analysed for specimen thicknesses of 12.5, 25, and

50mm. All data sets follow the Master Curve prediction accurately; however significant

ductile tearing occurred in 12 specimens of thickness 12.5mm, providing fracture

toughness values violating the size requirement (equation 28).

The size effect is clearly visible in this temperature category, in accordance with

weakest link behaviour.


Figure 5-7: Failure Probability Diagram for 12.5, 25, and 50mm specimens at T= -40oC


All thicknesses were tested at -20oC, and the non-size adjusted results are

presented in Figure 5-8.

Only three cleavage points are relating to the Master Curve in the first diagram

for 12.5 mm thick specimens, and there is significant ductile tearing in most test

specimens with a significant number of specimens exceeding the size requirement. The

number of non-failures (invalid points) is also rather considerable for 25mm specimens,

as loss of constraint and ductile tearing becomes significant above the M=30 toughness

criterion. The 100mm thick specimens indicate a higher Kmin value, which can be

estimated via other methods [50], [61].


Figure 5-8: Failure Probability Diagram for 12.5, 25, 50, and 100mm specimens at T= -20oC


Only 25mm thick test specimens were tested at -10oC, to assess upper shelf

behaviour. Four out of five specimens exhibited ductile tearing, by breaching the

0.05(W-ao) or 1mm stable crack growth criteria, and the KJ limit line. As only one valid

data point were obtained, the master curve interpretation of the results holds very little

validity.

Figure 5-9: Failure Probability Diagram for 25mm thick specimens at T= -10oC


5.1.8 Analysis at T= 0oC

All four specimen sizes were tested at 0oC, and the non size adjusted results can

be seen in Figure 5-10 below. For the 12.5mm and 25mm specimens, only one data

point is valid to the Master Curve and it is likely there is blunting at the crack tip. The

master curve interpretation of the results holds very little validity in this case, due to a

single data point being applicable.


Figure 5-10: Failure Probability Diagram for 12.5, 25, 50, and 100mm specimens at T= 0oC

5.1.9 Analysis at T= +20oC

This was the highest test temperature, where the three largest specimen sizes

were tested. The 25mm and the 50mm specimens all had ductile crack extensions

greater that 0.05(W-ao) and so the Master Curve analysis was not applicable. The

100mm size specimens only had two valid data points, meaning that assigning a Kmin is

unreliable.

The tests demonstrate that the initiation of the cleavage fracture is even possible

at high KJc values, and high temperatures.


Figure 5-11: Failure Probability Diagram for 25, 50, and 100mm specimens at T= 20oC


5.1.10 Interim Conclusions

The Master Curve predicts an increase in the lower bound of cleavage fracture

toughness with increasing temperature.

Secondly, the scatter associated with cleavage fracture toughness is observed to

increase with increasing temperature.

Thirdly, at high temperatures, failure by cleavage is preceded by ductile tearing, the

amount of which is also observed to increase as the test temperature increases.

Finally, the influence of specimen size on fracture toughness is observed to be more

marked at higher test temperatures. Larger specimens will have a higher transition

temperature, due to higher triaxial stress at the crack-tip, due to high constraint, than a

smaller specimen of the same material. Taken together, these observations are

consistent with a weakest link mechanism for cleavage initiation that is controlled by a

critical stress and likely to be affected by the constraint of the test specimen and any

pre-cleavage tearing[9].


5.2 Materials Characterisation

5.2.1 Optical Microscopy

Optical micrographs of the Euro RPV material observed in the L-S, S-T and L-T

orientations are shown in Figure 5-12 to Figure 5-14.

Figure 5-12: Microstructure of the A508 Cl.2 in the L-S plane of a RPV segment

Figure 5-13: Microstructure of the A508 Cl.2 in the S-T plane of a RPV segment


Figure 5-14: Microstructure of the A508 Cl.2 in the L-T plane of a RPV segment

The micrographs reveal the Euro material to have a ferrite-pearlite microstructure

with grain sizes of approximately 7 to 8 microns in diameter when viewed from all

directions. The size was determined using the linear intercept measurement

technique[62]. This indicates that the material has an equiaxed grain structure. The

revealed grain size is typical of acicular ferrite that contains a smaller grain size (~1-

10μm) than pearlite (~10-50μm), seen in Figure 2-2. This would most likely be due to a

faster cooling rate, and bypassing the pearlite phase region. This upper bainitic

microstructure forms at around 400-550oC in "sheaves" of ferrite plates [6] seen in the

micrographs here.


5.2.2 Scanning Electron Microscopy

Analysis of the microstructure of the Euro material, in particular the carbide

distribution, is shown in the SEM image in Figure 5-15. As shown from both optical and

SEM microscopy, ferritic steels are inhomogeneous with regards to the sub structure of

individual grains. The SEM images show that a number of coarse carbides are formed

in the bainitic ferrite matrix, and that grains and grain boundaries both contain carbides

that could act as nucleation sites for cleavage microcracks or ductile voids.

In can be seen that fine needle-type carbides are widely distributed, and appear

to be approximately 2μm in length. Kim et al [10] identifies these to be of the M2C type,

that have high Mo content and a hexagonal structure. The coarse carbides may be of

the M3C type, as investigated by Kim et al on A508 steel, which have high Fe content

and an orthorhombic structure. Further analysis will be required to confirm these

observations for this material.

Figure 5-15: SEM image of A508 Cl.2


5.2.3 Transmission Electron Microscopy

Figure 5-16 is a TEM image of the A508 Cl.2 carbon extraction replica. The

smaller grain boundaries in particular can be seen, as well as the coarse carbides, and

fine needle-type carbides. This image is at higher magnification than the SEM image,

revealing even smaller carbides on the nano scale.

Figure 5-16: TEM image of A508 Cl.2


Carbides within the matrix of the material are hard brittle particles that when

sampled within the fracture process zone near a crack tip, can initiate a micro crack that

can propagate with a cleavage fracture. To break bonds in the material microstructure,

the concentration of local stress must be sufficiently high to prevail over the cohesive

strength of the material. This is largely dependent on the temperature of the material.

The material characterisation of the A508 Cl.2 shows small grain sizes typical of


acicular ferrite. This microstructure is advantageous because its chaotic ordering

increases toughness[6].

The brittle carbides randomly distributed in the matrix decrease the fracture

toughness of the material however, and are present within grains and on grain

boundaries.

5.3 Finite Element Analysis

5.3.1 Preliminary investigation of crack interaction for double crack

specimens

All models were validated against published solutions both elastically and elastic-

plastically. Finite element models were validated elastically by comparing the KI

solution at a given load with that provided by Murakami[63] for an edge cracked

specimen under pure bending. The elastic stress field ahead of the crack tip was also

compared with that predicted by the Westergaard solution[26].

Elastic-plastic behavior was validated by comparing stress contours in the

vicinity of the crack tip from the FE results and that presented in the literature[47].

Finally, the limit-load derived from the models were validated against the relevant Miller

solution[64].

5.3.1.1 Elastic Validation

The stress-intensity factor for a cracked plate under four point bend is given below[63]:

5-28

Where:

P= Load

L= (S1-S2)/2


B= Thickness

W= Width

a= Crack length

A summary of the FE analysis KI solution and the Murakami solution can be seen

in Table 5-1. The FE analyses show good agreement with the standard KI solution

given by equation 6-1, with agreement between the solutions being close to 1% or less.

This suggests that the mesh is suitably refined at the crack tip for elastic analyses, and

that elastic properties have been correctly defined.

SEN(4PB) Model Load Applied

(kN)

FE analysis

KI solution

(MPa√m)

Murakami

KI solution

(MPa√m)

Difference

(%)

a/W = 0.5

Single crack 28 83 84 0.5

a/W = 0.5

Offset cracks 22 66 66 0.5

a/W = 0.1

Single crack 55 51 52 1.2

a/W = 0.1

Offset cracks 44 41 41 1.0

Table 5-1: Summary of Elastic validations of 2D FE models against Murakami KI Solution

The stress fields ahead of the crack tip were also validated against the

Westergaard functions[26] using equation 3-8, for mode I loading for a given applied KI.

A value of θ = 0o was used to analyse stress fields directly ahead of the crack. This was

repeated for all crack sizes, and an example of this can be seen below for the single

crack model of a/W = 0.5. All FE models showed good correlation with the elastic crack

tip stress field solution and with no deviation, even at very close distances to the crack

tip.


Table 5-2: Crack tip stress field elastic validation against Westergaard solution

5.3.1.2 Elastic-Plastic Validation

Stress contour plots from SENB(4PB) models with a/W = 0.5 were compared

against the contour plots published by Anderson and Dodds [47] for the same value of

a/W.

As a three point bend specimen was analysed by Anderson and Dodds[47], this

was also modeled in the FE software, with the same geometry depicted in Figure 4-1.

S1 = 190mm remained as for the four-point bend specimens, but the top pin was

centered at the symmetry plane for the three point bend model.

These results are illustrated in Figure 5-17. The non-dimensional contours of

maximum principal stress of 3.5 σy, 3.2 σy and 2.8 σy are presented at a low level of

loading. Three specimens of a/W = 0.5 are presented:

a) Three point bending,

b) Four point bending and

c) Double cracked specimen in four point bending.


The equivalent stress contours for SENB specimens with the same a/W ratio and

same material properties are shown in Figure 5-18 by Anderson and Dodds. The

principal stress contours of the SSY solution match both the shape and magnitude of

those for all specimens, which is expected at this level of J.

As the principal stress contours from Figure 5-17 and Figure 5-18 match in both

shape and magnitude, it is known that the SEN(B) stress contours in Figure 5-17 are

under a SSY condition. This is the case for the double crack specimen as well as the

single crack specimen. They are highly constrained with stresses and strains increasing

proportionally in accordance with the HRR singular field, with magnitudes governed by

the single parameter J.

Figure 5-19 shows maximum principal stress contours in the shallow cracked

SEN(B) specimens a/W = 0.1, for a low level of loading, at 3.5 σy, 3.2 σy and 2.8 σy. The

effect of the lower constraint condition on normalised principal stress contours is

significant, causing a large reduction in the area bounded by each contour when

compared with the high constraint condition described above and illustrated in Figure

5-17 and Figure 5-18 The reduction in the stress contours when normalized by J,

demonstrates that the magnitude of the opening mode stress is no longer governed by

the single parameter J.

Figure 5-20 shows a comparison of maximum principal stress contour shapes for

σ1 = 3 σy, under an increasing applied crack driving force (J). A progressive decrease in

the area bounded by the normalised stress contour can be seen. This is the effect of

constraint loss. Although the normalised stress contour size can be seen to decrease,

the absolute size actually increases with J, but not at the same rate predicted by the

SSY field.


Figure 5-17: Max Principal Stress Contours in SEN(B) specimens a/W = 0.5, n = 10

Figure 5-18: Comparison of Principal stress contours in small-scale yielding and in SENB specimen for a/W

= 0.5, n = 10 [47]


Figure 5-19: Max Principal Stress Contours in SEN(B) specimens a/W = 0.1, n = 10

Figure 5-20: Comparison of stress contours of Maximum Principal Stress, σ1 = 3 σy, for increasing

levels of loading, n = 10. Shallow crack (left) and Deep crack (right)

-5

-4

-3

-2

-1

0

1

2

3

4

5

-2 -1 0 1 2 3 4 5 rσy/

J

rσy/J

J=2.2 kN/mm2

J=3.6 kN/mm2

J=14.2 kN/mm2

J= 62.0 kN/mm2

-5

-4

-3

-2

-1

0

1

2

3

4

5

-2 -1 0 1 2 3 4 5 rσy/

J

rσy/J

J=26.7 kN/mm2

J=41.1 kN/mm2

J=101.5 kN/mm2

J= 230.6 kN/mm2


5.3.1.3 Load-Displacement Behaviour

The predicted limit load behavior for all specimen geometries with single and

double cracks have been validated against the Miller [64] limit load solution, for an

SENB in pure bending. The Miller solution defines the limiting moment, from which the

limit load can be defined. The limit loads calculated for the deep cracks and shallow

cracks are 22.4 kN and 65.8 kN respectively. This is irrespective of single or double

crack configurations, as the bending moment is the same within the same S2 pin

distance for each specimen[65].

The predicted load and CMOD behavior of the deep crack specimens can be

seen below in Figure 5-21. Using the ASME twice the elastic slope method[66], the limit

load provided by the Miller solution is conservative when compared with the FE

analysis, with a difference within 5%.

The load - CMOD behavior for the shallow crack specimens can be seen in

Figure 5-22. The FE analysis also predicted a limit load with a difference within 7% with

the solution provided by Miller[64].

Figure 5-21: Relationship between the applied force and CMOD in four point bend specimens, a/W = 0.5

0

5

10

15

20

25

30

35

40

0.00 0.30 0.60 0.90 1.20 1.50 1.80 2.10 2.40

Forc

e (

kN)

CMOD (mm)

Model 7.10.1 (a/W= 0.5, single crack)

Model 8.5.1 (a/W=0.5, offset cracks)


Figure 5-22: Relationship between the applied force and CMOD in four point bend specimens, a/W = 0.1

5.3.1.4 Crack tip stress fields

Crack tip stress fields were plotted to compare the stress concentration in

material ahead of the crack tip with the highly constrained HRR stress field, and to

evaluate the magnitude of this difference field, known as the Q parameter [43, 44]. The

numerical parameters required to calculate the HRR stress field in a plane strain

analysis with hardening exponent n = 10 were obtained from a report by Shih [67].

Figure 5-23 and Figure 5-24 show the crack tip stress fields at varying crack

driving forces for the single crack geometry and double crack specimen, respectively.

There is good correlation between the two specimens, with agreement between

comparative stress fields within 5%. This suggests that the cracks in the double

specimen behave independent of each other.

Figure 5-25 and Figure 5-26 also show good correlation between the principle

stress fields, suggesting shallow cracks in the double- cracked specimen behave

independently.

0

10

20

30

40

50

60

70

80

0.00 0.05 0.10 0.15 0.20

Forc

e (

kN)

CMOD (mm)

Model 9.4.1 (a/W= 0.1, Single crack)

Model 10.6.0 (a/W= 0.1, offset crack)


Figure 5-23: Normalised opening- mode stress fields ahead of crack tip (a/W = 0.5, single crack)

Figure 5-24: Normalised opening-mode stress fields ahead of crack tip (a/W = 0.5, offset cracks)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

0 1 2 3 4 5 6

σyy

/σ0

rσy/J

HRR Stress Field

(a/W= 0.5, single crack, J= 13.1 kN/mm2





0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

0 1 2 3 4 5 6

σyy

/σ0

rσy/J

HRR Stress Field

(a/W= 0.5, double crack, J= 16.6 kN/mm2

(a/W= 0.5, Double, J= 57.1 kN/mm2

(a/W= 0.5, Double crack, J= 118.5 kN/mm2




Figure 5-25: Normalised opening-mode stress fields ahead of crack tip (a/W = 0.1, single crack)

Figure 5-26: Normalised opening-mode stress fields ahead of crack tip (a/W = 0.1, offset cracks)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

0 1 2 3 4 5 6

σyy

/σ0

rσy/J

HRR Stress Field

(a/W= 0.1, Double crack), J= 7.6 kN/mm2




0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

0 1 2 3 4 5 6

σyy

/σ0

rσy/J

HRR Stress Field






5.3.1.5 J-Q Toughness Loci Relationship

Figure 5-27 and Figure 5-28 shows the variation of the crack driving force, J as a

function of constraint, quantified by Q. The value of Q becomes more negative as the

crack driving force, J, increases. The normalised opening-mode stress fields ahead of

the crack tip are used to define Q, with HRR as the reference field between the

reference stress and the opening-mode stress at rσy/J = 2. Q becomes more negative in

the a/W = 0.1 specimens than the a/W = 0.5 specimens, signifying that there is a

greater loss of constraint on the crack tip stress fields in the small crack geometry,

compare Figure 5-27 and Figure 5-28.

Both plots compare within 10% to numerical modeling results reported in the

literature[39] for a material with similar stress strain behaviour.

Figure 5-27: J-Q Toughness Locus for a/W = 0.5 specimens


Figure 5-28: J-Q Toughness Locus for a/W = 0.1 specimens

5.3.1.6 Crack driving force versus load

The variation of elastic plastic crack driving force as a function of load is shown

in Figure 5-29 for both shallow and deeply-cracked specimens containing single or

double cracks. There is reasonable agreement between the data for single and double

cracks for both deeply cracked and shallow cracked specimens. For deeply cracked

specimens, the maximum difference in crack driving force is approximately 50MPa√m

at the end of the analyses. Within the crack driving force range of interest, 200 to

300MPa√m, the difference is approximately half this value, i.e. approximately 10%. For

the shallow cracked specimens, agreement is better, with less than 2% difference

between the single crack and offset crack models.


Figure 5-29 KJc vs Load for all specimens

5.3.2 Sensitivity analysis of testing specimens with different pre-crack

lengths

Finite element models were validated elastically and elastic-plastically in a

similar approach taken for the preliminary 2D analysis. As described in greater detail

previously in section 5.2.1, models were validated:

1. Elastically, by comparing the KI solution at a given load with the stress intensity

factor solution provided by Murakami [63].

2. Elastic-plastically by comparing the limit load from load displacement behaviour

against the Miller[64] limit load solution for an SENB in pure bending.

5.3.2.1 Elastic Validation

Table 5-3 shows the level of agreement between the FE analyses KI solutions

and the associated Murakami solution. J values were taken from the centre of the

0

50

100

150

200

250

300

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140

KJ MPa√m

Load, (kN)

Model 7.10.1 (a/W= 0.5, single crack)

Model 8.5.1 (a/W=0.5, offset cracks)

Model 9.7.3 (a/W= 0.1, Single crack)

Model 10.7.1 (a/W= 0.1, offset crack)


specimen. The table shows very good agreement for the a/W = 0.55 cracks for each of

the three test options, with agreement between the solutions to be 5% or less.

SEN(4PB) Model Load Applied

(kN)

FE analysis

KI solution

(MPa√m)

Murakami

KI solution

(MPa√m)

Difference

(%)

a/W = 0.55

Offset Cracks 10 67 67 0.5

a/W = 0.55

One on top of the other 13 80 83 3

a/W = 0.55

Next to each other 12 72 76 5

Table 5-3: Summary of Elastic validations against Murakami KI Solution for a/W = 0.55

5.3.2.2 Load - Displacement Behaviour

The load - CMOD behaviour for all test configurations are shown in Figure 5-30

to Figure 5-43 to understand the effect that the various test arrangements have.

For comparison with 3D FEA data, 2D models of SEN(4PB) specimens were

created for further analysis. This consisted of a single crack specimen with a/W = 0.55

and another single crack specimen with a/W = 0.45. The overall geometry of the

specimens was consistent with the 3D model geometry, seen previously in Figure 4-4.

The 2D models were created to show the effect of using plane strain and plane stress

elements on the global force - CMOD behaviour. As expected, the plane strain

assumption requires much greater force for the same displacement to occur. As a real

specimen will have both plane stress and plane strain regions at the crack tip, we

expect the 3D FEA models to have force - CMOD behaviour between the contrasting

plane strain and stress variants. The 2D analysis with plane strain and stress elements

are present on all figures in this section as a reference to the material behaviour seen in

the 3D models.

In Figure 5-30 The behaviour of a real specimen is represented by the 3D model

of a single crack SEN(4PB) that can be seen in the image of the figure. This behaviour


is due to both plane strain and plane stress regions at the crack front, and is to be

expected. The basic four-point bend geometry was modelled to see if the other test

arrangements produced similar material behaviour to this.

To validate the elastic compliance of the plane strain and stress behaviour, the

elastic compliance solution provided by Saxena and Hudak [68] is provided by the

dotted straight lines. Both plane stress and strain elastic compliances agree to the

literature value within 3%.

Figure 5-31 Shows a separate 3D model that has the same geometry and

material properties as the specimen in Figure 5-30, but with a/W = 0.45. The 2D models

in this figure have the same crack length to width ratio for comparison. The difference in

force - CMOD behaviour for the a/W = 0.45 specimen compared to the 0.55 is that the

slightly shallower crack shows stiffer behaviour.

Figure 5-30: Force - CMOD for a single crack SEN(4PB) specimen, a/W = 0.55.

2D plane stress and plain strain with elastic compliance solutions are also plotted.

0

2

4

6

8

10

12

14

16

18

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Fo

rce (

kN

)

CMOD (mm)

Elastic compliance Plain Strain (A.Saxena and S.J Hudak)

Elastic compliance Plain Stress (A.Saxena and S.J Hudak)

2D Plain Strain aW=0.55, Single crack

2D Plain Stress aW=0.55, Single cracks

3D Quarter model 0.55 a/W Single Crack


Figure 5-31: Force - CMOD for a single crack SEN(4PB) specimen, a/W = 0.45.

2D plane stress and plane strain with elastic compliance solutions are also plotted.

0

2

4

6

8

10

12

14

16

18

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Fo

rce (

kN

)

CMOD (mm)

Elastic compliance Plain Strain (A.Saxena and S.J Hudak)

Elastic compliance Plain Stress (A.Saxena and S.J Hudak)


2D Plain Stress aW=0.45, Single crack

(Model 22.0.0) 3D Quarter model 0.45 a/W Single Crack


Two SEN(4PB) specimens loaded in parallel.

Figure 5-32 and Figure 5-33 show the force - CMOD behaviour of two specimens

with different pre crack lengths mounted in four point bending in a parallel configuration.

As with the previous figures, the 2D analyses with both plane stress and strain

elements are seen in grey.

The sensitivity in force - CMOD behaviour of testing two specimens with slightly

different pre-crack lengths become paramount when testing specimens in parallel. This

is because in the actual fracture mechanics test, only a single load cell can be used to

measure the load on both specimens; however the load will not be distributed evenly

between the specimens due to the different compliances of the specimens. The deeper

crack specimen is more compliant to the load, and hence will experience less of the

force being applied. This can be seen in Figure 5-32. The yellow curve represents the

applied force if it were split equally between the two specimens, which would be the

case if the pre-crack lengths where the same. The red curve represents the corrected

force - CMOD behaviour, which unfortunately cannot be measured in an actual test, but

by FEA analysis only. Figure 5-33b shows a similar scenario, but in this case, the

specimen will receive more of the shared load due to being less compliant. The red

curve corrects for this, hence displays the actual behaviour, which is stiffer than when

equally distributing the load, seen by the orange curve.

Sensitivity of testing cracks with slightly different pre-crack lengths becomes

more of an issue when testing two specimens under the same loading condition in this

manner. To obtain accurate force - CMOD results, extensive FEA analysis with

accurate measurements of the pre-crack lengths would have to be conducted of every

test scenario. Other disadvantages of testing this method rather than an offset crack

specimen, is that more material will be needed to produce two specimens per test

rather than one, and that the specimens will have to be further machined, to ensure

they contact the rollers equally.


Figure 5-32: Force - CMOD for single crack a/W = 0.55 SEN(4PB) specimen loaded in parallel. 2D plane stress

and plane strain also plotted.

Figure 5-33: Force - CMOD for single crack a/W = 0.45 SEN(4PB) specimen loaded in parallel.

2D plane stress and plane strain also plotted.

0

2

4

6

8

10

12

14

16

18

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Fo

rce (

kN

)

CMOD (mm)


2D Plain Stress aW=0.55, Single cracks

aW=0.55, In Parallel

aW=0.55, In Parallel, Correct Force

0

2

4

6

8

10

12

14

16

18

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Fo

rce (

kN

)

CMOD (mm)



aW=0.45, In Parallel



Two SEN(4PB) specimens loaded in series.

Figure 5-34 and Figure 5-35 show the force - CMOD behaviour of two specimens

with different pre crack lengths mounted in four-point bending on top of each other, i.e.

in series. As with the previous figures, the 2D analyses with both plane stress and strain

elements are included as grey lines.

Both figures show the force - CMOD behaviour as expected, in comparison with

the 2D models. This is an advantage over the previous loading method, as each

specimen will receive the load that is measured via the load cell, and the associated

CMOD can be read with individual clip gauges. Difficulties may arise in the actual test

with frictional effects affecting results. This is due to the inner rollers that are in contact

with both specimens, and each specimen will cause the rollers to move in opposing

directions. The crack driving force could also be limited by the diameter of the inner

rollers. More material will be required than testing a single specimen with offset cracks,

and experimental challenges such as alignment of specimens could pose an issue.

Misalignment of specimens could result in mixed loading of the specimens, and

possibility of both specimens cleaving due to the initial cleavage event may occur.

The sensitivity in force - CMOD behaviour of testing two specimens with slightly

different pre crack lengths is also of importance when testing specimens in series. This

can be seen from crack driving force versus load data shown in

Figure 5-36. Even with sufficiently high loads of 18kN, the crack driving forcse for the

slightly shallower cracked specimen ceases to increase, whilst that experienced by the

deeper cracked specimen increases to much higher stress intensities. This is due to the

effects of shielding [69].

Sensitivity of testing cracks with slightly different pre-crack lengths is an issue

when testing two specimens under the same loading condition in this manner. Pre-

crack lengths must be of exceptionally close similarity to each other for both specimens

to propagate to the desired stress intensity at temperatures near the onset of upper

shelf temperature. If there is a significant difference in pre-crack length, then the

controlling fracture parameter will depend on the deeper cracked specimen, rather than

its microstructural constituents, which are not desired.


Figure 5-34: Force - CMOD for single crack a/W = 0.55 SEN(4PB) specimen loaded in series.2D plane stress

and plane strain also plotted.

Figure 5-35: Force - CMOD for single crack a/W = 0.45 SEN(4PB) specimen loaded in parallel.


0

2

4

6

8

10

12

14

16

18

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Fo

rce (

kN

)

CMOD (mm)


2D Plain Stress aW=0.55, Offset cracks

aW=0.55, In series

0

2

4

6

8

10

12

14

16

18

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Fo

rce (

kN

)

CMOD (mm)



aW=0.45, In series


Figure 5-36: Crack driving force versus load for single crack SEN(4PB) specimens, a/W = 0.55 and 0.45

loaded in series.

A single SEN(4PB) specimen with offset cracks

Figure 5-37 and Figure 5-38 show the force - CMOD behaviour of a single

specimen, which has offset cracks with different pre crack lengths mounted in four point

bending. The 2D analyses with both plane stress and strain elements are seen in grey.

Both figures show the force - CMOD behaviour as expected, in comparison with

the 2D models. One advantage of this test method rather than loading the specimens in

parallel and in series is that it requires half the amount of material than the other test

methods and setting up the test is more straightforward.

As each crack is placed between the inner rollers, the bending moment on each

crack is the same; hence the same load will be applied to each crack. The difference in

CMOD can be easily quantified with individual clip gauges. The FEA results for this

specimen show very little sensitivity in force - CMOD behaviour of testing two cracks

with slightly different pre crack lengths. Both cracks show insignificant interaction with

0

100

200

300

400

500

600

700

0 5 10 15 20

KJ (M

Pa√

m)

Load (kN)

aW=0.55, In series

aW=0.45, In series


each other, as the behaviour is comparable to other models, and also there is less

sensitivity from different pre-crack lengths compared to other test arrangements.

Figure 5-37: Force - CMOD for a single SEN(4PB)specimen with offset cracks. Plot shown is for a/W = 0.55.


0

2

4

6

8

10

12

14

16

18

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Fo

rce (

kN

)

CMOD (mm)


2D Plain Stress aW=0.55, Offset cracks

aW=0.55, Offset cracks


Figure 5-38: Force - CMOD for a single SEN(4PB)specimen with offset cracks. Plot shown is for a/W = 0.45.


Summary all three test arrangements

Figure 5-39 and Figure 5-40 show all the force - CMOD plots of the FEA analysis

conducted for the sensitivity study of testing specimens with differing pre-crack lengths.

In comparison to the green curve (a single cracked specimen in four point

bending), all plots show very similar force - CMOD behaviour. The offset crack

geometry has a slightly less stiff elastic slope in both a/w = 0.55 and 0.45 cracks.

Figure 5-40 shows stiffer force - CMOD behaviour, which is to be expected as

the crack length is slightly shorter.

In all cases, the force - CMOD behaviour are largely consistent with each other,

which is to be expected due to the consistency in pin spacing, materials properties and

specimen geometry.

Taking into account the advantages and disadvantages of each test method, the

3D analyses have shown that the test approach with a single specimen with offset

0

2

4

6

8

10

12

14

16

18

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Fo

rce (

kN

)

CMOD (mm)





cracks is the most favourable. This is due to the efficiency of material usage, and the

simplistic nature of the test setup, in comparison to the other methods. The most

important advantage of this test method is that the load applied to each crack can be

quantified, without detailed FE analysis, which is a big flaw with the 'in parallel' method.

Its major advantage over the 'in series' method is that the effects of shielding will not

affect the stress intensity that the cracks will experience, and the amount of deformation

is not attributed to the diameter of the inner rollers. There are also less frictional affects

to be concerned with, as all rollers in the offset crack test set up can freely rotate.

Figure 5-39: Applied force and CMOD for all SEN(4PB) specimens, a/W = 0.55.


0

2

4

6

8

10

12

14

16

18

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Fo

rce (

kN

)

CMOD (mm)





aW=0.55, In series



Figure 5-40: Applied force and CMOD for all SEN(4PB) specimens, a/W = 0.45.


Force- CMOD of a/W 0.5 and 0.1 Offset cracks specimens

Due to the advantages of the offset cracked specimen design in comparison to

the other specimen arrangements, two further offset crack models where created to

compare force - CMOD behaviour. These were an offset deeply cracked specimen with

a/W = 0.5, and a very shallow offset cracked specimen, of a/W = 0.1 seen in Figure

5-41 and Figure 5-42 respectively. The force - CMOD behaviour, Figure 5-43, shows

that a much greater force must be applied to the shallow crack specimen than the

deeply cracked specimen for the same crack mouth opening displacement to occur.

This is to be expected due to the increase in remaining ligament length in the shallow

crack specimen.

0

2

4

6

8

10

12

14

16

18

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Fo

rce (

kN

)

CMOD (mm)





aW=0.45, In series



Figure 5-41: Offset cracked SEN(4PB) specimen, a/W = 0.5

Figure 5-42: Offset cracked SEN(4PB) specimen, a/W = 0.1

Figure 5-43: Applied force and CMOD for SEN(4PB) specimens with offset cracks, a/W = 0.5 and another

specimen a/W = 0.1

0

5

10

15

20

25

30

35

40

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Fo

rce (

kN

)

CMOD (mm)

(Model 23.0.1) 3D Quarter model 0.5 a/W Offset cracks



5.3.2.3 Crack driving force versus load

The variation in crack driving force as a function of load for all test arrangements that

have a/W = 0.55 is shown in Figure 5-44. The 2D FEA analyses with plane strain and

plain stress elements are also plotted as a reference. A 3D single crack SEN(4PB)

specimen with a/W = 0.55 is plotted to further predict the behaviour of the more

complex test geometries. All three test arrangements match the behaviour of the single

crack specimen well, within 5% accuracy. Demonstrating that all test arrangements

could generate accurate results.

Figure 5-45 has a similar approach taken, with 2D models that had a plane strain and

plane stress analyses, plotted as a reference. Also plotted is a simplified test

arrangement of a 3D single cracked specimen loaded under four point bending with

a/W = 0.45. As with the deeper cracks, all three test arrangements match the behaviour

of the simplified test accurately to within 5%.

The main observations with the data are the shortcomings of some of the test

geometries. For the "in parallel" specimens, the corrected load is shown which can be

post processed from the FEA. As noted previously, in the real test, the actual force on

each specimen cannot be determined. Only one load cell can be installed on to the test

rig, and equal sharing of the load would be an over estimation of the load being applied

to the deeper crack, and an underestimate of the load being applied to the shallower

crack. For the "in series" specimens, the deeper crack specimen experiences higher

stress intensities as the load increases, but this does not occur with the shallower crack

specimen.

Figure 5-45 shows the specimen with a/W = 0.45 only experiencing a stress intensity of

approximately 200MPa√m, where as the deeper crack will continue to deform under

stress intensities of 600 MPa√m and more, as Figure 5-36 showed previously . The

specimen with the offset cracks produce KJ vs Load data similar to the other analyses

undertaken, showing that there is insignificant crack interaction between the two cracks

at the fracture toughness that occurs at the OUST, 300MPa√m. Also, FEA analyses

show that pre-cracks of slightly different length in an offset crack specimen do not affect


results as much as the other test methods with different pre-crack lengths.

Figure 5-44 Crack driving force vs Load behaviour for cracks with a/W = 0.55 for all test arrangements.

Figure 5-45: Crack driving force vs Load behaviour for cracks with a/W = 0.45 for all test arrangements.

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20

KJ (M

Pa√

m)

Load (kN)



(Model 21.0.0) 3D Quarter model 0.55 a/W Single Crack aW=0.55, Offset cracks

aW=0.55, In series


0

50

100

150

200

250

300

350

400

450

0 5 10 15 20

KJ (M

Pa√

m)

Load (kN)



(Model 22.0.0) 3D Quarter model 0.45 a/W Single Crack


aW=0.45, In series



Crack driving force vs load for specimens with a/W 0.5 and 0.1 Offset cracks

Due to the preference of testing two cracks under the same loading conditions by

an offset crack specimen, the crack driving force versus load behaviour has been

plotted for the two additional models. To achieve any level of crack driving force, a

much greater force is required in the specimen with the shallow offset cracks than the

deeper crack specimen. This is expected due to the larger uncracked ligament in the

shallow cracked specimen. The behaviour seen in Figure 5-46 is therefore to be

expected, and within the crack driving force range of interest, 200 to 300 MPa√m, the

deeply cracked specimen requires a load of approximately 15 to 17 kN. This differs

significantly in the shallow cracked specimen, requiring a load of approximately 45 to

53 kN.

Figure 5-46: Crack driving force vs Load behaviour for specimens with offset cracks. One specimen of

a/W = 0.5 and another of a/W = 0.1

0

50

100

150

200

250

300

350

400

0 5 10 15 20 25 30 35 40 45 50 55 60 65

KJ (M

Pa√

m)

Load (kN)




5.3.2.4 Crack tip stress contours

In order to analyse the crack tip constraint condition at the onset of upper shelf

temperature, the crack tip stress contours of maximum principal stress have been

plotted for all three test variants using data from Abaqus software[1]. The stress

contours in Figure 5-47 relate to a low level of loading, of approximately J = 17kN/m2.

The results shown are normalised against the yield stress (414 MPa) of the material

and the applied crack driving force. At this level of crack driving force, and that the

normalised contours are of equivalent shape and magnitude, it is known that these

contours are under SSY conditions. These highly constrained contours have stresses

and strains increasing proportionally in accordance with the HRR singular field, and are

governed by the crack driving force, J, only.

All three test arrangements show comparable stress contours within 5% of each

other. This is to be expected due to the same specimen geometry, crack length and

spacing of the rollers.

Due to the favourable test method of offset crack specimens due to factors

mentioned in the previous section, the maximum principal stress contours at a low level

of loading for an offset crack specimen with a/W = 0.5 have been analysed in Figure

5-48. These contours of high stress are bound by a slightly reduced area than the

contours in Figure 5-47, which is attributed to the slightly shorter crack length. Also,

these contours are under a SSY condition.

The effect that crack tip constraint loss has on the maximum principal stress

contours can be seen from Figure 5-49, which are the normalised contours of the same

stress levels as the previous contour plots, at 3.5 σy, 3.2 σy and 2.8 σy. This reduction in

area bounded by each contour is due to the constraint loss arising from the geometrical

change in crack length. This specimen has a/W = 0.1, resulting in a 2.5mm crack

length. The reduction in area when normalized by J demonstrates that J no longer

governs the magnitude of opening mode stress, but the effects of constraint loss are

much more contributing factor to the fracture toughness of specimens with small cracks.


Figure 5-47: Maximum Principal Stress Contours in SEN(4PB) specimens in three different test

arrangements, a/W = 0.55, n=10

Figure 5-48: Maximum Principal Stress Contours in SEN(4PB) specimen that has offset cracks,

a/W = 0.5, n=10

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

-1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

rσy/

J

rσy/J

(4PB a/W 0.55) Offset Crack 3.5σy



(4PB a/W 0.55) In Series 3.5σy



(4PB a/W 0.55) In Parallel 3.5σy



-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

-1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

rσy/

J

rσy/J






a/W = 0.1, n=10

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

-1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

rσy/

J

rσy/J





Comparison of opening mode stress contours for increasing levels of loading

To observe the effect that increased loading has on maximum principal stress

contours, a stress value of 3σY was chosen, and the stress contour was plotted over

three increasing levels of loading. The cracks were loaded to crack driving forces of

approximately 200, 300 and 400 MPa√m, which equates to the J values seen in Figure

5-50. These values were selected because the estimated fracture toughness for RPV

steels near the OUST occurs between 200 and 300 MPa√m.

All three test arrangements are included in this analysis, and show similar results

with less that 5% difference. Due to normalising the contours with J, a progressive

decrease in the area bounded by each contour can be seen. The stress contour that

results from 200 MPa√m of crack driving force appears the largest; however the actual

contour size would be small in comparison to higher loading levels. The shape of the

stress contour also changes with increased loading. At approximately 400 MPa√m, the

contours for all tests show a more pronounced curvature in contour shape at 0o to the

crack tip.

At the crack driving forces present in the analyses, Figure 5-51 shows very

similar shape and magnitude to the equivalent contours present in the previous figure.

Figure 5-51 shows the contours from an offset cracked specimen, with a/W = 0.5. This

specimen has a slightly smaller crack length than the specimens in Figure 5-50, but

there is no effect in contour area at these high levels of loading.

Figure 5-52 shows maximum principal stress contours at increasing levels of

load for an offset specimen with a very small crack, a/W = 0.1. The contour size is

considerably smaller in this analyse than that observed in the previous analyses

performed in respect of the deeply cracked specimens. This indicates that significant

constraint loss occurs throughout the loading process, and to achieve the same stress

contour area as a deeper crack, a much higher crack driving force needs to be applied.


Figure 5-50: Maximum Principal Stress Contours, σ1 = 3σY for increasing levels of loading in SEN(4PB)

specimens. Three different test arrangements shown, a/W = 0.55, n=10

-3.5

-2.5

-1.5

-0.5

0.5

1.5

2.5

3.5

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

rσy/

J

rσy/J

J= 185.4 kN/mm2 Offset Cracks aW=0.55



J= 171.6 kN/mm2 In series aW=0.55

J= 484 kN/mm2 In series aW=0.55

J= 757.9 kN/mm2 In series aW=0.55

J= 179.0 kN/mm2 In Parallel aW=0.55





specimen that has offset cracks, a/W = 0.5, n=10


specimen that has offset cracks, a/W = 0.1, n=10

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

rσy/

J

rσy/J




-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

rσy/

J

rσy/J





Analyses of crack tip stress contours at actual size

Figure 5-53 shows the actual sizes of the contours from Figure 5-47. These are

not normalised with respect to J, hence their true sizes can be seen. Although each

crack is under a nominally SSY condition, it is clear from this figure that the actual

stress contour size varies from specimen to specimen. This is due to the actual applied

crack driving force, J, applied in each case. The "in parallel" specimens (shown in red)

had the lowest applied J in this analysis of 15.4 kN/m2, the offset crack specimen (in

blue) had J = 17.6 kN/m2 and the "in series" specimens (in yellow) had the highest

applied J = 21.0 kN/m2. This is apparent in the analysis as the contours from each test

arrangement increase in that order.

Figure 5-54 shows the actual contour sizes from Figure 5-48. The applied J in

this analysis was 16.8 kN/m2. As the crack tip acts as a stress concentrator, we can see

the region of highest stress, 3.5σy, occurs at close distances from the crack tip. This

stress contour is bounded within 0.07mm at 0° to the crack tip.

Figure 5-55 shows the actual contour sizes from Figure 5-49. These contour

sizes are caused by a crack driving force of J = 18.7 kN/m2. Due to constraint loss at

the crack tip, the region of highest stress is bound by a very small contour area, and

occurs within 0.02mm at 0° to the crack tip.


Figure 5-53: Maximum Principal Stress Contours in SEN(4PB) specimens in three different test

arrangements, a/W = 0.55, n = 10 (Actual size)


a/W = 0.5, n=10 (Actual size)

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

r (m

m)

r (mm)










-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

r (m

m)

r (mm)






a/W = 0.1, n=10 (Actual size)

Contour size matching

Due to crack tip constraint loss, the shallow crack specimen will have smaller

contours of maximum principal stress at a given level of crack driving force than the

deeper crack specimen, as seen previously. Figure 5-56 below shows the increase in

crack driving force required on the shallow crack specimen to produce stress contours

of a similar area to that observed in the deeply crack specimen. When the deep crack

specimen of a/W = 0.5 is in SSY conditions, at a low level of loading, its contour size

reaches up to 0.3mm away from the crack tip at J = 16.8 kN/m2 (or KJ = 60.8 MPa√m).

The load required for this can be found from Figure 5-46 to be approximately 10kN. For

the shallow specimen of a/W = 0.1, a much higher crack driving force of

J = 117.6 kN/m2 (or KJ = 160.8 MPa√m) is required to produce contours of similar area.

The load for this from Figure 5-46 is approximately 41kN, which is a significant increase

in load compared to the deeply cracked specimen.

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

r (m

m)

r (mm)





Figure 5-56: Contour matching of offset crack specimen, a/W = 0.5 at a low level of loading,

and an offset crack specimen, a/W = 0.1 at a higher crack driving force

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

r (m

m)

r (mm)




J= 117.6 kN/mm2 (4PB a/W 0.1) Offset Crack 3.5σy




Actual size contours at KJ = 250 MPa√ m

Figure 5-57 shows the actual contour size in the deeply cracked specimen at a

crack driving force of 250 MPa√m, the approximate predicted fracture toughness at

OUST. At OUST, there is 5% probability of cleavage preceded by 0.2 mm ductile

tearing[56]. At this temperature, the shallow crack specimen did not produce contours

of equivalent size, which is the critical size for onset of upper shelf behaviour to occur.

This signifies that the critical stress state at the crack tip for cleavage initiation does not

occur, and that the shallow crack specimen is unlikely to cleave. For onset of upper

shelf temperature to occur, the shallow crack specimen would need to be tested at a

lower test temperature, and design FEA analysis must therefore be conducted using

tensile properties relevant to a lower temperature. This can be seen from the schematic

in Figure 5-58. At the size of contour of interest, (red dotted line) the crack driving force

of the deep crack specimen is 250 MPa√m, but the shallow crack specimen never

reaches this size of contour, inhibiting cleavage initiation. Only ductile tearing will

prevail, unless the yield stress is lowered by testing at a lower temperature.

Figure 5-57: Actual size contours at a stress intensity of 300 MPa√m, The predicted fracture toughness at

OUST.

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

r (m

m)

r (mm)





Figure 5-58: Schematic of how crack tip stress contour area varies with crack length at increasing fracture

toughness.


Offset crack specimens behave independently to each other when loaded in four

point bending, at the separation distances explored in these analyses. This is because

the central section of the specimen between the inner loading rollers has a constant

bending moment, resulting from the symmetry of the S1 pins in relation to the S2 pins.

As each double crack will each incur the same bending moment as a single crack, then

each crack will act independently of each other. This is true for high constraint

geometries as it is for low constraint geometries of a/W = 0.1.

It is noteworthy that the change in crack tip stress field increases from the HRR

stress field at increasing normalized distances in the deeper crack specimens (Figure

5-23 and Figure 5-24), i.e. the Q parameter is not constant with increasing distance

from the crack tip. This is not the case for short crack specimens (Figure 5-25 and

Figure 5-26) where the displacement from HRR stress field remains consistent after a

normalized distance of rσy/J = 1. This is due to crack tip fields impinging on the global

0

50

100

150

200

250

300

350 F

ractu

re T

ou

gh

ne

ss, K

MA

T (

MP

a√m

)

Area of stress contour

Shallow

crack

Deep

crack

SSY

Behaviour


bending field in a/W = 0.5 specimens. Crack tip stress fields will initially scale with J in

accordance to SSY, and independent of geometry, but as load increases, this bending

component will become more significant. The low constraint geometries have lower

stress fields than the high constraint geometries, and this is reflected with the greater

negative values in Q parameter for a given applied J, seen in Figure 5-27 and Figure

5-28. This is consistent with published work by Nevalainen and Dodds [39].

The sensitivity analysis of testing specimens with different pre crack lengths

show that specimens loaded "in parallel" will not distribute the applied load evenly

between the specimens of slightly different pre crack lengths. The deeper crack is more

compliant to the load, and hence will experience less of the load applied in comparison

to the shallow crack that will experience more of the load. As only one load cell is used,

estimating that the load is evenly split between the two specimens will not provide

accurate results.

The specimens loaded "in series" may have alignment issues due to the

unstable nature of the test set up, and also frictional effects regarding the inner two

rollers may compromise accuracy in the results. FEA analysis also shows that a slightly

shallower cracked specimen may cease propagation, whilst the deeper cracked

specimen proceeds to much higher stresses. This is due to the effects of shielding.

A specimen with offset cracks may present pre-cracking difficulties, however if

sufficiently spaced apart (such as 55mm, in the analyses) then they can be pre-cracked

individually via three point bending. This will avoid any undue crack driving force on the

neighbouring crack. The stress contours of all three test arrangements have noticeable

similarity, within 5% difference.


6.0 Summary and Conclusions

This thesis has presented the results of research aimed at developing a new

methodology with which to assess the combined influence of constraint, microstructure

and fracture mechanism on the fracture toughness behaviour of RPV steels at

temperatures near the OUST. The main summary conclusions from the work are as

follows:

1. Statistical analyses of Euro RPV ferritic steel material fracture toughness

properties over the temperature range -154°C to +20°C have been performed using the

Master Curve approach. The analyses have shown that:

1.1. The Master Curve predicts an increase in the lower bound of cleavage fracture

toughness, defined with respect to a 5% cleavage probability, with increasing

temperature.

1.2. The scatter associated with cleavage fracture toughness is predicted to increase

with increasing temperature.

1.3. At high temperatures failure by cleavage is preceded by ductile tearing, the

amount of which is also to increase as the test temperature increases.

1.4. The influence of specimen size on fracture toughness is observed to be more

marked at higher test temperatures. Larger specimens exhibit a higher transition

temperature compared with smaller specimens.

2. Metallographic examination of the Euro RPV steel has been performed using optical,

scanning electron and transmission electron microscopy. The examination has shown

that:

2.1 The Euro material exhibits equiaxed ferrite pearlite microstructure with a grain

size of between 7 and 8µm. This is an upper bainitic microstructure that forms in

"sheaves" of ferrite plates.

2.2 Fine needle-like carbides are observed within the matrix and at the grain

boundaries of the material.


3. The stress contour-based approach of Anderson and Dodds [47] has been explored

as an approach for studying the influence of constraint on the Onset of Upper Shelf

behaviour using finite element analysis. The results have shown that:

3.1 Contours of maximum principal stress plotted in the vicinity of the crack-tip

provide a useful approach for assessing constraint effects on cleavage

fracture over the transition temperature regime; the area bounded by a

particular contour, e.g. 3.5 y, provides an indication of constraint level.

4. Finite element analysis, combined with the stress contour approach, has been used

to assess a number of testing approaches to study the mechanism of fracture near the

onset of upper shelf temperature. The following conclusions can be made:

4.1 A single specimen containing two offset cracks is the most favourable test

approach for studying the influence of prior ductile damage and constraint on

the onset of upper shelf temperature when compared with loading two

specimens either in series of in parallel. This is because:

o specimens in series will experience frictional effects due to the rollers

being unable to rotate, and there will be a limit to the maximum applied

displacement; and

o specimens loaded in parallel with slightly different crack lengths will have

different compliances. This means that individual forces on each

specimen cannot be characterised without detailed analysis.

4.2 Analyses of offset cracked specimens containing deep and shallow cracks

have demonstrated that in the crack driving force range of interest:

o each crack behaves in a similar manner to cracks in a fracture mechanics

specimen containing a single crack, and therefore

o each crack behaves independently of each other.


7.0 References

[1] Hibbit, Karlsson, and Sorenson, “ABAQUS Standard v6.10 Finite Element Software.” Dassault Systemes, 2010.

[2] A. H. Sherry, “Special Issue on Cleavage,” Fatigue Fracture of Engineering Materials and Structures, vol. 29, no. 29, pp. 659–660, Sep. 2006.

[3] T. L. Anderson, Fracture Mechanics- Fundamentals and Applications, 3rd Editio. .

[4] V. . Shah and P. . MacDonald, “Pressurized Water Reactor Pressure Vessels,” 1993.

[5] G. R. Odette and G. E. Lucas, “Embrittlement of Nuclear Reactor Pressure Vessels,” JOM, 2001. [Online]. Available: http://www.tms.org/pubs/journals/JOM/0107/Odette-0107.html#ToC2.

[6] H. K. D. H. Bhadeshia and R. W. K. Honeycombe, Steels: Microstructure and properties. Butterworth-Heinemann, 2006.

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