Critical Size of a Postulated Crack in the Irradiated Nozzle-

Vessel Junction of the Pressurized Water Reactor for the

Reactor Vessel Head Drop Accident Impact Load

by

Waleed Ishaque

A project submitted in conformity with the requirements

for the degree of Masters in Engineering

Mechanical and Industrial Engineering Department

University of Toronto

Copyright by Waleed Ishaque 2015

ii

Critical Size of a Postulated Crack in the Irradiated Nozzle-Vessel

Junction of the Pressurized Water Reactor for the Reactor Vessel Head

Drop Accident Impact Load

Waleed Ishaque

Masters of Engineering

Mechanical and Industrial Engineering Department

University of Toronto

2015

Abstract

Critical size of a postulated external surface semi-elliptical circumferential crack in the three loop

Combustion Engineering pressurized water reactor nozzle-vessel junction is calculated using

ANSYS Workbench (Academic version) built-in crack object with the applied impact load from

the vessel head drop accident. Failure assessment diagram for numerous crack depths and lengths

was developed taking into account the fracture toughness properties of the irradiated reactor vessel

steel. The mode I stress intensity results used in the failure assessment diagram were compared

with available finite-element and API 579 analytical solutions for purposes of validation, showing

good agreement.

iii

Acknowledgments

I am in debt to Professor Jan Spelt for the tremendous support he provided me during the course

of this project. His technical guidance especially during numerous challenges kept me going. He

was a mentor in my professional and academic career. I am grateful to have him as my supervisor

because I have grown under his guidance. He shared with me ideas that taught me about the art of

engineering and seeing complex problems from the basic principles. Thank you Professor Jan

Spelt, I will always remember this.

I would also like to thank Dr. Aman Usmani for lending me his time when I first met him eagerly

to discuss a project for my Masters of Engineering degree. He provided guidance and suggested

industry project papers that led me to the topic of my project. Thank you for giving me a start.

Thanks to Dr. Waleed Mekky and Dr. Soliman Soliman for the motivation and support through

the past year. They both supported me through the ups and downs as I juggled part-time studies

with full-time stress analysis role at AMEC NSS.

Last but not the least; I would like to thank my family of non-engineers for listening to me patiently

when I talked at the dinner table of the little things I learned about fracture mechanics.

iv

Table of Contents

Acknowledgments .......................................................................................................................... iii

Table of Contents ........................................................................................................................... iv

List of Tables ................................................................................................................................. vi

List of Figures ............................................................................................................................... vii

Background and Problem Definition ...................................................................... 1

1.1 Introduction ......................................................................................................................... 1

1.2 Pressurized Water Reactor .................................................................................................. 2

1.3 Neutron Embrittlement ....................................................................................................... 2

1.4 Reactor vessel head drop accident ...................................................................................... 4

1.5 Summary ............................................................................................................................. 8

Literature Review .................................................................................................... 9

2.1 Introduction ......................................................................................................................... 9

2.2 Linear Elastic Fracture Mechanics ...................................................................................... 9

2.2.1 Griffiths crack growth criteria ............................................................................... 9

2.2.2 Stress Intensity Approach: Griffiths Modified Theory by Irwin ......................... 11

2.3 Elastic Plastic Fracture Mechanics ................................................................................... 14

2.3.1 J-Integral ............................................................................................................... 14

2.3.2 Differences and Relationship between LEFM and EPFM .................................... 15

2.4 Available Solutions for Nozzle-Vessel Junction External Circumferential Surface

Cracks ............................................................................................................................... 18

2.5 Fracture toughness ............................................................................................................ 26

2.5.1 Master curve method ............................................................................................. 26

2.5.2 Unified Curve Method .......................................................................................... 27

v

2.5.3 Irradiated RPV steel fracture toughness ............................................................... 28

2.6 Fracture Assessment Diagram .......................................................................................... 29

2.7 Summary ........................................................................................................................... 33

Scope and Methodology ....................................................................................... 34

3.1 Introduction ....................................................................................................................... 34

3.2 Scope ................................................................................................................................. 34

3.3 Methodology ..................................................................................................................... 37

3.3.1 Assumptions .......................................................................................................... 39

3.3.2 Phase I Finite Element Analysis Validation ...................................................... 40

3.3.3 Phase II API579 Solution ................................................................................... 43

3.3.4 Phase III Critical Flaw Size Assessment ........................................................... 45

3.4 Summary ........................................................................................................................... 48

Results and Discussion ......................................................................................... 49

4.1 Introduction ....................................................................................................................... 49

4.2 Phase I ............................................................................................................................... 49

4.2.1 Stress intensity evaluation for crack due to tension and shear .............................. 49

4.2.2 Stress intensity evaluation for crack due to bending ............................................ 54

4.3 Phase II .............................................................................................................................. 58

4.4 Phase III ............................................................................................................................ 61

4.5 Summary ........................................................................................................................... 62

References ..................................................................................................................................... 66

vi

List of Tables

Table 1: Differences between Linear-Elastic and Elastic-Plastic Fracture Mechanics ................................................ 15

Table 2: API 579 reference paragraphs for component, crack and loading conditions ............................................... 22

Table 3: API 579 Influence parameter Go ................................................................................................................... 23

Table 4: API 579 Influence parameter G1 ................................................................................................................... 23

Table 5: Pressurized water reactor vessel head drop impact load by W. C. Castillo et al. (2009) ............................... 45

Table 6: Fracture assessment diagram data of the irradiated A533 nozzle vessel junction flaw ................................. 62

Table 7: Fracture assessment diagram data of the new A533 nozzle vessel junction flaw .......................................... 62

vii

List of Figures

Figure 1: Illustration of the Three Loop Combustion Engineering pressurized water weactor vessel .......................... 3

Figure 2: Illustration of the pressurized water reactor vessel resting on the bracket support ........................................ 1

Figure 3: Illustration of neutron embrittlement mechanics (G.R. Odette and G.E. Lucas, 2001).................................. 3

Figure 4: Illustration of a reactor assembly ................................................................................................................... 5

Figure 5: Illustration of a reactor disassembly............................................................................................................... 5

Figure 6: Illustration of a reactor reassembly ................................................................................................................ 5

Figure 7: Illustration of reactor restart ........................................................................................................................... 5

Figure 8: Illustration of reactor vessel head drop impact reaction load and load location ............................................. 6

Figure 9: ANSYS model for the postulated crack in the nozzle-vessel junction ........................................................... 7

Figure 10: Illustration of the API 579 nozzle and crack geometry (section view) ...................................................... 19

Figure 11: Illustration of the API 579 nozzle and crack geometry (front view) .......................................................... 19

Figure 12: Illustration of the nozzle-vessel junction crack profiles from API 579 ...................................................... 21

Figure 13: Unified curve for the fracture toughness of the irradiated and new A533 steel ......................................... 28

Figure 14: Illustration of the failure assessment diagram ............................................................................................ 33

Figure 15: CAD drawing of the three loop Combustion Engineering pressure vessel reactor .................................... 35

Figure 16: Finite-element model of the three loop Combustion Engineering reactor outlet nozzle ............................ 36

Figure 17: Finite-element of the Phase I nozzle-vessel junction model ...................................................................... 40

Figure 18: Illustration of the Phase I methodology ..................................................................................................... 41

Figure 19: Illustration of the crack tip parameters from K. Sedighiani, 2011 ............................................................. 42

Figure 20: Finite-element of the Phase II nozzle-vessel junction model ..................................................................... 43

Figure 21: Illustration of the Phase II methodology .................................................................................................... 44

Figure 22: Finite-element of the Phase III nozzle-vessel junction model .................................................................... 46

file:///C:/Users/Waleed/Dropbox/M.Eng%20project/Working%20folder/Report/M.%20Eng%20project%202Feb2015.docx%23_Toc410682859file:///C:/Users/Waleed/Dropbox/M.Eng%20project/Working%20folder/Report/M.%20Eng%20project%202Feb2015.docx%23_Toc410682862

viii

Figure 23: Illustration of the Phase III methodology ................................................................................................... 47

Figure 24: Stress intensity due to Fy for the Phase I nozzle-vessel junction crack with a/c=1/3 and a/t=0.2 .............. 51

Figure 25: Stress intensity due to Fz for the Phase I nozzle-vessel junction crack with a/c=1/3 and a/t=0.2 .............. 52

Figure 26: Stress intensity due to Fx for the Phase I nozzle-vessel junction for a crack with a/c=1/3 and a/t=0.2 ...... 53

Figure 27: Stress intensity due to Mx for the Phase I nozzle-vessel junction for a crack with a/c=1/3 and a/t=0.2 .... 55

Figure 28: Stress intensity due to My for the Phase I nozzle-vessel junction for a crack with a/c=1/3 and a/t=0.2 .... 56

Figure 29: Stress intensity due to Mz for the Phase I nozzle-vessel junction for a crack with a/c=1/3 and a/t=0.2 ..... 57

Figure 30: Polynomial fit to the stress profile of the Phase II nozzle-vessel junction ................................................. 59

Figure 31: ANSYS and API 579 mode I stress intensity for the Phase II nozzle-vessel junction ............................... 60

Figure 32: Fracture assessment diagram of the Phase III irradiated A533 nozzle-vessel junction .............................. 64

Figure 33: Fracture assessment diagram of the Phase III non-irradiated (new) A533 nozzle-vessel junction ............ 65

Figure 34: Neutron shield tank RPV support system (W.G. Hopkins., 1987) ............................................................. 68

Figure 35: Cantilever type RPV support system (W.G. Hopkins., 1987) .................................................................... 69

Figure 36: Column type RPV support (W.G. Hopkins., 1987) .................................................................................... 70

Figure 37: Bracket type RPV support (W.G. Hopkins., 1987) .................................................................................... 71

Figure 38: General arrangement drawing of the Westinghouse PWR ......................................................................... 72

Figure 39: Upper vessel assembly of the Westinghouse PWR .................................................................................... 73

Figure 40: Upper vessel machining of the Westinghouse PWR .................................................................................. 74

Figure 41: Inlet nozzle of the Westinghouse PWR ...................................................................................................... 75

Figure 42: Inlet nozzle cladding and machining of the Westinghouse PWR .............................................................. 76

Figure 43: Outlet nozzle of the Westinghouse PWR ................................................................................................... 77

Figure 44: Outlet nozzle cladding and machining of the Westinghouse PWR ............................................................ 78

Figure 45: Typical fuel handling arrangement (Westinghouse, 1984) .......................................................................... 1

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ix

1

Background and Problem Definition

1.1 Introduction

The objective of this project was to perform an assessment of the postulated scenario where a crack

is present in the critical stress region of the nozzle-vessel junction during the pressurized water

reactor vessel head drop event. The reactor vessel geometry selected for this project was the Three

loop Combustion Engineering pressurized water reactor since its drawings were available in the

public domain and it shares a similar geometry used in the head drop event analysis by D.W.

Alexander et al. (1978) and W.C. Castillo et al. (2009).

The impact loads from the pressurized water reactor vessel head drop event (W.C. Castillo et al.,

2009) were used in the analysis. The mode I stress intensity results for the postulated external

surface circumferential semi-elliptical crack in the nozzle-vessel region were compared to the

fracture toughness properties of the irradiated steel to represent the current aging fleet of reactors

in the United States.

The fracture assessment diagram technique was used to integrate the latest available fracture

toughness properties and the stresses in the nozzle-vessel junction from the head drop event loads

to determine a critical size of the postulated crack. The motivation for considering this approach

stems from the aging fleet of reactors that are operating with irradiated metal with reduced fracture

toughness. The critical size of the postulated flaw may provide an upper limit of assurance that

after the vessel head drop event, the structural integrity of an aged reactor power vessel is

maintained. Therefore, the ultimate goal of this project was to determine the critical size of the

postulated external surface semi-elliptical circumferential on the nozzle-vessel junction of the

pressurized water reactor vessel with the head drop event impact load conditions. Fulfilling the

objective of the project required knowledge of (1) the pressurized water reactor vessel structure,

(2) fracture toughness properties of the irradiated reactor pressure vessel steel and (3) fracture

mechanics techniques. Each topic is briefly discussed in the next sections.

2

1.2 Pressurized Water Reactor

The pressurize water reactor (PWR) is the most common type of reactor used in the United States.

In a PWR, the reactor pressure vessel (RPV) is the largest single component which is exposed to

radiation. One of the primary functions of an RPV is to contain the reactor core and its radiation.

It also houses numerous safety systems that cool and control the nuclear fuel. The RPV consists

of a cylindrical vessel with a removable semi-hemispherical head for access to the reactor core and

nozzles for coolant flow (E.G. Hopkins et al., 1987). A computer-aided illustration of the three

loop Combustion Engineering PWR, the selected reactor vessel geometry for this project, is shown

in Figure 1.

3

Figure 1: Illustration of the Three Loop Combustion Engineering pressurized water weactor vessel

The RPV is structurally supported in four different ways (each shown in Appendix A): (1) Neutron

shield tank support, (2) Column support, (3) Cantilever support and (4) Bracket support. The

bracket support is the most relevant to this project and is shown in Figure 2 as an illustration.

Nozzle

Guide studs

Reactor vessel

head

Reactor vessel

1

Figure 2: Illustration of the pressurized water reactor vessel resting on the bracket support

Nozzle support

bracket

Concrete

cavity

2

1.3 Neutron Embrittlement

There are two nuclear fission by-products found inside the reactor core of a pressurized water

reactor: radiation and neutrons. Radiation causes ionization but neutrons, more specially the fast

neutrons with energy greater than 1 MeV, cause structural damage to metals by knocking out the

lattice atoms to a new location in the lattice structure. As these atoms leave their original location

and travel, defects like vacancies and interstitials are formed. Also, a special molecular behavior

occurs where the copper atoms precipitate to form what is known as the Copper Rich precipitate

(CRP) which contribute directly to the hardening of the steel by pinning dislocations. CRPs is just

one of the many steel hardening factors. Other complex "matrix-features" also take part in the

process and are currently the subject of research (E.D. Eason et. al., 2006). Defects act as obstacles

and enhance the ability of lattice structure to deform under load.

Defect formation has recently been analyzed using molecular dynamics, Monte Carlo simulations

and experimental measurements. While it is emphasized here that the process of neutron

embrittlement is complex, defect formation originally described by G.R. Odette and G.E. Lucas

(2001) is summarized as follows:

High energy neutron impacts an atom producing highly energetic primary recoiling

atoms (PRA)

The PRA energy is transferred to nearby atoms creating vacancies and self-interstitials

Most of the self-interstitial atoms recombine with vacancies. Atoms that do not

recombine form "matrix-features" which are produced due to precipitation of solute-

atoms such as copper

Defects and obstacles (for example copper rich precipitates) formed during energy

precipitation process increase steel hardening (neutron embrittlement) by pinning the

lattice structure dislocation

3

Figure 3: Illustration of neutron embrittlement mechanics (G.R. Odette and G.E. Lucas, 2001)

Since the high energy neutrons are the leading cause of embrittlement, measuring a materials

exposure to these neutrons is key in performing assessment of an aging reactor component. A

standard unit of measurement is the neutron flux which is the rate of the neutron exposure over

a unit area per unit time (neutrons/m2 s). Neutron exposure over a period of time is called the

neutron fluence (neutrons/m2). A United States pressurized water reactor is typically designed

to operate for 30-40 years with a neutron fluence of 1x1023 to 3x1023 neutrons/m2s depending on

the specific design (G.R. Odette and G.E. Lucas, 2001).

4

1.4 Reactor vessel head drop accident

A pressurized water reactor has to be refueled at scheduled intervals for continued operation. The

entire fueling/refueling operation can be described in four phases:

Phase 1 preparation: In this phase the reactor is shut down and is kept in a subcritical

state in preparation for the refueling operation. Figure 4 is a simplified illustration of

the PWR pressure vessel in relation to the reactor building where it is resting inside a

concrete cavity.

Phase 2 reactor disassembly: As shown in Figure 6, the reactor vessel head is lifted

by the polar crane inside the reactor building. During the lift, the concrete cavity is

filled with water simultaneously as shown in Figure 5. Once the vertical lift operation

is complete, the vessel head is stored on a pedestal (pedestal not shown in Figure 5)

while the concrete cavity is kept filled with water during the entire refueling operation.

Phase 3 reactor assembly: After the reactor core fuel is replaced with fresh fuel, the

vessel head is lowered. Once the vessel head is engaged to the guide studs at

approximately 14 feet from the vessel flange, the water is completely drained while the

vessel head remains elevated to allow for inspections of the guide studs. This

configuration is similar to the one as shown in Figure 6.

Phase 4 reactor restart: In this final stage the reactor is fully reassembled and is

prepared for start-up as shown in Figure 4 and Figure 7.

During Phase 2 and 3, there is a risk that the elevated vessel head might drop on the vessel flange.

If the vessel head drops on the flange, the reactor outlet and inlet nozzle-vessel junction will

experience the highest bending stress due to the impact load. Using the maximum reaction loads

from a recent head drop sensitivity analysis performed by W.C. Castillo et al (2009), the stress

analysis and flaw assessment performed in this project determined the critical size of the postulated

external surface semi-elliptical crack at the nozzle junction with the fracture toughness of the

irradiated vessel at the refueling operation temperature of approximately 38 oC (I. Namgung et al,

2005). The ANSYS model built and used for the analysis is shown in Figure 9.

5

Figure 4: Illustration of a reactor assembly

Figure 5: Illustration of a reactor disassembly

Figure 6: Illustration of a reactor reassembly

Figure 7: Illustration of reactor restart

Reactor

building

Main

pump

Polar

crane

Concrete

cavity Boiler

Reactor

pressure

vessel

Guide

studs Vessel

head

Crane

hook

Water

6

Figure 8: Illustration of reactor vessel head drop impact reaction load and load location

Impact load

on nozzles

7

Figure 9: ANSYS model for the postulated crack in the nozzle-vessel junction

8

1.5 Summary

This chapter introduced the objective of this project: to determine the critical size of the postulated

flaw (external surface semi-elliptical circumferential crack) on the nozzle-vessel junction of a

pressurized water reactor vessel with the head drop event impact load conditions. By integrating

mode I stress intensities with the irradiated metal fracture toughness properties, the critical size of

the postulated flaw at the nozzle-vessel junction of a pressurized water reactor vessel is determined

and discussed in the last chapter of this project report. A technical background on the necessary

theoretical concepts and application tools used in the project is presented in the next chapter in

form of literature review.

9

Literature Review

2.1 Introduction

A literature review was conducted to examine (1) the historical background of fracture mechanics,

(2) the existing methodologies to analyze nozzle-vessel junction external surface circumferential

semi-elliptical cracks, (3) the fracture toughness modelling of irradiated metals and finally (4) the

fitness-for-service evaluation procedures. Each of the items are discussed at suitable lengths to

support the analysis that was performed as part of this project.

2.2 Linear Elastic Fracture Mechanics

2.2.1 Griffiths crack growth criteria

M. Janssen , J. Zuidema and R.J.H. Wanhil (2002) provide a historical account of the development

of Fracture Mechanics since its demonstration as a phenomena for the first time in A. A. Griffiths

pioneering paper "The Phenomena of Rupture and Flow in Solids" (1920):

"The breaking load of a thin plate of glass having in it a

sufficiently long straight crack normal to the applied stress, is

inversely proportional to the square root of the length of the

crack. The maximum tensile stress in the corners of the crack is

more than ten times as great as the tensile strength of the

material, as measured in an ordinary test."

What today is known as Griffith's theorem is the representation of an energy balance between a

through-thickness crack in a uniformly loaded infinite plate. The Griffith's crack growth criteria

suggests that a crack of length 2 will grow when the release of elastic energy, , in a body due

to an incremental growth of crack, (2), is higher than the surface energy required, , to

10

sustain that same crack growth. Therefore, an energy balance for total energy, , of the plate, can

be shown as

(2)=

(2)

+

(2)< 0 Eq. (1)

The energy balance above is for the case without external work being performed on the plate. To

arrive with a useful relationship the energy balance is expanded. Assuming that by introducing a

crack in a plate we have essentially eliminated or "released" the uniform applied stress, , around

crack parameter, the energy released, , per unit thickness can then be written as

=

= 0.5

= 2 0.5

= 22

Eq. (2)

To then define surface energy, , for a unit thickness, it is simply the relationship of crack length

and surface energy of the material i.e. surface tension, , as shown in Eq. (3).

= 2 2

= 4

Eq. (3)

By substituting equation Eq. (3) and Eq. (2) in Eq. (1) the energy balance per unit thickness of the

total plate energy is

(2)=

(2)

+

(2)< 0

=

(2)( + ) < 0

Eq. (4)

11

=

(2)(

22

+ 4) < 0

Eq. (4) is Griffiths crack growth criteria and can also be presented as

2

> 2

>2

Eq. (5)

Since the right hand side of Eq. (5) is a material constant; it indicates that the energy balance

becomes unstable when the value of exceeds a certain critical value. This critical value was

later quantified by Dr. George Rankine Irwin as the critical stress intensity .

2.2.2 Stress Intensity Approach: Griffiths Modified Theory by Irwin

With much credit to Dr. Griffith for his work, the utility of fracture mechanics took birth with Dr.

George Rankin Irwin's modification to Griffith's theorem. Dr. Irwin's modification comes with

designation of left hand side of equation Eq. (5) as the energy release rate , , and the right hand

side as crack resistance, .

2

> 2

>

Eq. (6)

To derive in terms of energy, Eq. (4) for a crack in center of a plate is rewritten as

=

(2) Eq. (7)

And for an edge crack at the side of a plate

12

=

Eq. (8)

To make Griffiths energy approach useful, Dr. Irwin developed an equation for the stress field

around the crack tip

=

2() + Eq. (9)

where , are the cylindrical polar co-ordinates of a point in relation to the crack tip and is the

ratio of the stress field to the applied stress also known as the stress intensity. It was then also

shown that the stress intensity factor in its general form is

= (

) Eq. (10)

where is the remotely applied stress, (/) is an influence parameter which depends on the

crack geometry and specimen width . If (/) = 1 for a case of infinite plate with a central

crack with length, 2, then for plane stress

=2

=

2

Eq. (11)

and for plane strain

=

2(1 2)

Eq. (12)

Therefore, combining the energy approach with the stress intensity; the parameter governing the

fracture of a specimen is the critical energy release rate, , which may be difficult to directly

13

measure from an experiment. But experiments can be conducted to measure the critical stress

intensity which is directly related to .

The widely used equation = > in introductory Linear Elastic Fracture Mechanics

(LEFM) is rooted from Dr. Griffith's pioneering research. What makes LEFM so powerful is our

ability to experimentally determine the value for the critical stress intensity . Numerous amounts

of research has since been conducted to develop some of the most commonly used stress intensity

factor equations. On the other hand, LEFM is based on an assumption that the plastic zone around

the crack tip remains small in comparison with the crack length. But as the engineering application

of materials became complicated the need for accounting for plasticity increased to obtain better

estimates. Modern fracture mechanics problems have recently been treated using Elastic-Plastic

Fracture Mechanics (EPFM) which however is not a well-developed theory but is complemented

with modern engineering tools of Finite Element Analysis when complicated geometries are

encountered.

14

2.3 Elastic Plastic Fracture Mechanics

When the plastic zone at the crack tip is too large, LEFM assumptions are violated. In power

generation industry where thick wall pressure vessels are common, cracks are usually found in

high constraint situations where the fracture behavior is still controlled by highly localized yielding

instead of general net section yielding. To analyze failure with localized yielding, one of the most

common techniques used is the J-integral which was introduced by J.R. Rice in 1968.

2.3.1 J-Integral

The J-integral is the strain energy release rate on the surface of either side of a crack. The J-integral

found its way to the industry in 1981 by getting incorporated in ASTM E813 several years after

its introduction by J. R. Rice (1968). The J-integral is a theoretical concept based on line integral

but like everything else, is also related to the energy balance approach

= + + Eq. (13)

where is the total energy of a plate, is the change in elastic energy of the plate caused by

introducing a crack, is the change in surface energy of the plate caused by introducing a crack

and is the work performed by the force system during crack growth. The J-integral, , can then

be interpreted as the rate of change of elastic energy per unit between bodies on either side of a

crack with length

=

Eq. (14)

15

2.3.2 Differences and Relationship between LEFM and EPFM

The J-integral has an elastic, , and non-elastic component,

= + Eq. (15)

where the elastic component is related to the energy release rate

=

=

2

(1 2)

Eq. (16)

The plastic component of the J-integral is the measure of strain-energy release rate but it takes into

account the possibility of crack growth as the load increases The following table is a summary of

essential differences between the LEFM and EPFM analysis procedure at the most basic level:

Table 1: Differences between Linear-Elastic and Elastic-Plastic Fracture Mechanics

LEFM EPFM

The plastic zone is negligible, elastic

fracture is exhibited and and strain release rate criteria applies

, ,

The plastic zone is limited but not negligible, no-linear fracture is exhibited and

strain energy release rate criteria applies

<

16

LEFM EPFM

Based on the load vs. displacement curve

from a test with linear behaviour

demonstrated during crack growth:

Based on the load vs. displacement curve from

a test with non-linear behaviour demonstrated

during crack growth:

=

=

( + )

=

=

= +

= +

( )

Differences in LEFM and EPFM become apparent when the behavior of crack tip in each category

is observed. Based on Irwin's theory, it is known that the stresses tend to approach infinity at the

crack tip due to the so called 1/ singulatrity, where is the radius of crack. But as soon as the

sharp crack tip starts to open and become blunt, owing to plasticity in the region, the 1/

singularity is no longer valid. Instead, it has been shown the stresses are a function of 1/ which

is a finite value. Irwin (1967) calculates the size of the plastic zone size simply by

=

2

=

4

Eq. (17)

17

=1

2(

)

2

In modern fracture mechanics, American Standard of Testing Materials requires that the fracture

toughness measurement specimen meet the following

, > 2.5 (

)

2

Eq. (18)

where is the specimen thickness and is the crack length. In other words, since

=1

2(

)

2

Eq. (19)

then by rearranging and substituting Eq. (18)

< (1

2) (

a

2.5)