2-Fracture

Applied Metallurgy of Carbon Steels
An e-Oilfield Reference Guide
2 - Fracture
Peter Timmins, P.Eng.
External Examiner, APEGA
October 2020

CONTENTS
2
2 - Fracture
Peter Timmins
Basics – Stress and Strain
Engineering Stress and Strain – Tensile Test
Stress/Strain Curve
Some Common Material Models Widely Used in Fracture Mechanics
Definitions – Fracture and Fracture Mechanics
Objectives of Fracture Mechanics

Linear Elastic Fracture Mechanics (LEFM):
Stress Concentration -1
Liberty Ships -1
Liberty Ships -2
Comet Aircraft - Wells, A.A., ‘‘The Condition of Fast Fracture in Aluminum Alloys with Particular Reference to Comet Failures.” British
Welding Research Association Report, April 1955
Free Energy Change For Processes – e.g., Crack Growth and Atomic Separation, Nucleation and Growth Processes, etc.
Anderson’s Atomic View of Fracture
Griffith’s Energy Balance
Griffith’s Energy Balance and Irwin’s Strain Energy Release Rate
A.A. Griffith, “The Phenomena of Rupture and Flow in Solids”, Philosophical Transactions of the Royal Society of London, A 221,
163- 198 (1921) [ Presented February 26 1920].
Irwin, G.R., ‘‘Onset of Fast Crack Propagation in High Strength Steel and Aluminum Alloys.” Sagamore Research Conference
Proceedings,
Vol. 2, 1956, pp. 289–305.
Orowan - Irwin Modification of Griffith to Account for Plasticity
Fracture Toughness and Failure Regimes
Crack Resistance Curve (R curve) and Crack Driving Force Curve (G Curve).
K-G Relationship
Stress Intensity Factor and Crack Opening Modes
Crack Tip Singularities
Principal Stresses
Crack Tip Stress Fields
Stresses on the Crack Plane
Stress Intensity Factor (SIF)
Global Stress Fields
CONTENTS
3

4
CONTENTS
Stress Intensity Factor for Mode I Through Crack in an Infinite Plate
K for a Single Edge Cracked Specimen Mode I and Mode II Loading
KI Solutions for Common Test Specimens
KI Solutions for Common Crack Geometries
Crack Tip Plasticity - Irwin
Strip Yielding- Dugdale – Burdekin -1
Effective Crack Length
Strip Yielding - Limits
Physical Metallurgy of Crack Tip Plasticity SEM Measurements of SZW
Plane Stress and Plane Strain -1
Changes in Crack Direction: F. Erdogan and G. C. Sih, “On the Crack Extension in Plates Under Plane Loading and Transverse”
ASME, 1963
DHC in Zr-2.5wt.% Nb Compact Tension Specimens; Timmins, P., “Effect of Stress State and Texture on DHC Kinetics”,
ECF 9, Varna, Bulgaria, 21-25 September 1992
Plastic Zones – G.T Hahn and A.R. Rosenfield, “Local Yielding and Extension of a Crack Under Plane Stress”,
Washington D.C, 1964
Hall-Petch Relationship for Stress Intensity Factor, K: R.W. Armstrong, in: Eng. Fract. Mech., 28, [5-6], 529 (1987)
Linear Elastic Fracture Mechanics -Examples
Inclined Crack in Tension
Cylindrical Pressure Vessel with an Inclined Through-thickness Crack
Worked Example -1
Worked Example -1 - Solution
Worked Example -2
Worked Example -3

5
CONTENTS
Elastic-Plastic Fracture Mechanics (EPFM)
EPFM Regime – Crack Tip Plasticity
Contour Integral: Rice, J.R., “Some Remarks on Elastic Crack-Tip Stress Fields.” International Journal of Solids and Structures,
Vol. 8, 1972, pp. 751–758.: G.P. Cherepanov, “Mechanics of Brittle Fracture,” McGraw-Hill, New York (1979).
Example - Energy Release Rate by J-integral for a Double Cantilever Beam Specimen
Estimation of CTOD from the Strip Yield Model: Burdekin, F.M. and Stone, D.E.W., “The Crack Opening Displacement Approach to
Fracture Mechanics in Yielding Materials.” Journal of Strain Analysis,Vol. 1, 1966, pp. 145–153
J-CTOD Relationships
Crack-Growth Resistance Curves
JIc Measurement
J1.5mm -Prediction from Microstructure: Timmins, P. “Effect of Heat Treatment on the Ductile Fracture
Characteristics of Zr-2.5 wt.% Nb”, ECF 8, Turin, Italy, Oct 1-5, 1990
Fracture Mechanisms
Ductile Fracture
Smooth, Notched and Cracked Specimens Used in Ductile Fracture Studies
Phenomenological Modeling of Ductile Fracture in Steel
SEM Fractographs after Image Processing
Ductile Fracture: Void Nucleation, Growth, and Coalescence
Ductile Fracture in Round Notched Bars of High Strength Steel: Damage Accumulation, Initiation of Macroscopic Crack,
Crack Growth and Shear Lip Formation.
Slant Fracture
Zig-Zag Crack Growth by Void Nucleation, Growth, and Coalescence
More Ductile Crack Propagation by Void Growth and Coalescence
Modes of Void Coalescence – MnS in C-Mn Steels
Crack Growth by Void Nucleation, Growth, and Coalescence - Tunneling
DHC in Zr-2.5wt.% Nb Compact Tension Specimens; Timmins, P., “Effect of Stress State and Texture on DHC Kinetics”, ECF 9,
Varna, Bulgaria, 21-25 September 1992 - Tunneling
Ductile Crack Tunneling and Transition from Flat to Slant Fracture in Pre-cracked Specimens
Crack Growth Ahead of a Crack Tip by Void Nucleation, Growth, and Coalescence

6
CONTENTS
Cleavage Fracture
Dislocation Models of Cleavage Fracture
Comments on Dislocation Models of Cleavage Fracture
Smith’s Cracked Carbide Model (1966)
Fracture Initiation Ahead of Notch Root (J. F. Knott et alia) at Some “Characteristic Distance” at which Cleavage Fracture is “Triggered”
Ritchie-Knott-Rice Model for Cleavage Fracture - Characteristic Distance
Weakest Link
Weakest Link -2
Cleavage Trigger - Heerens, J. and Read, D.T., ‘‘Fracture Behavior of a Pressure Vessel Steel
in the Ductile-to-Brittle Transition Region.” NISTIR 88-3099, National Institute for Standards and
Technology, Boulder, CO, 1988.
SEM Fractograph of Cleavage in an A 508 Steel
Ductile-Brittle Transition
Ductility Transition from Void Coalescence to Cottrell Cracked Dislocation
Cleavage Crack Propagation in the Ductile-Brittle Transition Region.
Correlation of Tensile and Charpy Ductile-Brittle Transition Results: R.W. Armstrong,
G.R. Irwin and X.J. Zhang, in Cleavage Fracture; George R. Irwin Symposium (TMS-AIME, Warrendale, PA, 1998) pp. 51-58
Stroh, A. N. The Formation of Cracks as a Result of Plastic Flow, Proc. Roy. Soc. Lond., 223, 404-414, 1954.
Changes in Crack Direction: F. Erdogan and G. C. Sih, “On the Crack Extension in Plates Under Plane Loading and Transverse”
ASME, 1963.
Intergranular Fracture
Intergranular Fracture - Examples
Fatigue Crack Growth
Mechanism of Fatigue Crack Growth
Fatigue Crack Growth – Beach marks
EXAMPLE: Striation Width vs. da/dN

7
CONTENTS
Environmental Fracture
Mechanisms of Stress Corrosion Cracking and Hydrogen Embrittlement
Anodic Stress Corrosion Cracking- Mechanism
Cathodic Stress Corrosion Cracking- Hydrogen Embrittlement Mechanism
Fracture Toughness Testing
Overview and Evolution of Testing
Fracture Toughness, 𝐾𝐼c , Testing for Mode I - ASTM E399
J Testing : BS 7448 Part 1 and ASTM E 1820.
CTOD Testing
Fracture Toughness Testing of Weldments – Overview
Fracture Toughness Testing of Weldments – Notches in Specimens
Fracture Testing of Weldments – Post-Test Microstructures
Charpy and Izod Impact Testing - ASTM Standard E 23
Fatigue Crack Growth
Fatigue Crack Growth Rate - 1
Fatigue Crack Growth Models
Near Threshold Data for C-Mn Steel
Summary of Fatigue Crack Growth Regions
Defect Tolerance Analysis - (Time Dependent Flaw Growth)
Introduction to Failure Assessment Diagrams
The Strip-Yield Failure Assessment Diagram – CEGB
Welded Structures – Weld Mismatch FAD Example

8
CONTENTS
International FAD-Based Procedures for Fitness For Service Assessments of Structures Containing Crack-Like Flaws
Summary of International FAD-Based Procedures
Failure Assessment Diagram (FAD) – API 579 Methodology
Synthesis - API 579 and BS 7910
Example - Defect Classified as a Longitudinal Crack in a Pressurized C-Mn Steel Pipe
Solution - API 579
Solution – BS 7910
Solution – Synthesis API 579 BS 7910 - Concluding Remarks
Environment Assisted Cracking
Environment Assisted Cracking – SCC
Environment Assisted Cracking – Corrosion Fatigue
Environment Assisted Cracking - Summary of SCC and Corrosion Fatigue
FAD Approach to EAC - BS 7910
FAD Approach to EAC - TWI
Additional References

10
Engineering Stress and Strain – Tensile Test

11
UTS Fracture
Stress/Strain Curve

12
Some Common Material Models Widely Used in Fracture Mechanics

13
Fracture is the (local) separation of an object or material into two, or more,
pieces under the action of stress.
Fracture mechanics is the field of mechanics concerned with the study of
the propagation of cracks in materials. It uses methods of analytical solid
mechanics to calculate the driving force on a crack and those of experimental
solid mechanics to characterize the material's resistance to fracture
Definitions – Fracture and Fracture Mechanics

• What is the residual strength
as a function of crack size?
• What is the critical crack size?
• How long does it take for a
crack to grow from a certain
initial size to the critical size?
14
Objectives of Fracture Mechanics

Linear Elastic Fracture Mechanics
(LEFM)

16
Geometric discontinuities: holes, corners, notches, cracks etc, act as stress
concentrators/risers

17
Uniaxial Biaxial

Liberty Ship failures: WW2 +250 ships fractured or cracked;
19 of these broke completely in two.
18
Schenectady,
A liberty Tanker, 16 January 1943, split in two while
moored in calm water.
Only 24 hrs old
Esso Manhattan
29 March 1943 at the entrance to New York Harbor
Liberty Ships-1

Investigations revealed that the Liberty
ship failures were caused by a
combination of three factors:
• The welds, which were produced by a
semi-skilled work force, contained crack-
like flaws.
• Most of the fractures initiated on the
deck at square hatch corners, where there
was a
local stress concentration.
• The steel from which the Liberty ships
were made had poor toughness, as
measured by Charpy impact tests
19
Liberty Ships -2

• In 1956, A.A. Wells used
fracture mechanics to show that
the fuselage failures in several
Comet jet aircraft resulted from
fatigue cracks reaching a
critical size.
• These cracks initiated at
windows and were caused by
insufficient reinforcement
locally, combined with square
corners that produced a severe
stress concentration. (c.f., the
hatch design in the Liberty
ships.)
20
Comet Aircraft - Wells, A.A., ‘‘The Condition of Fast Fracture in
Aluminum Alloys with Particular Reference to Comet
Failures.” British Welding Research Association Report, April 1955.

Free Energy Change For Processes – e.g., Crack Growth and
Atomic Separation, Nucleation and Growth Processes, etc.
21
Crack Length
Critical Crack
Length
No Crack
Growth
Crack
Growth

Anderson defines the cohesive stress,
between atoms, σc, as:
Which is essentially the same as the
critical stress to cause crack propagation
as in the Griffith energy balance:
22
Potential energy and force as a function of atomic separation. At the equilibrium separation
Xo the potential energy is minimized, and the attractive and repelling forces are balanced.
Anderson’s Atomic View of Fracture

An incremental increase in crack area under
equilibrium conditions is given by:
or:
where:
This energy balance is shown opposite for
increase in crack area dA due to crack increment
𝑑a, and the creation of two new surfaces.
Potential energy in terms of strain energy, is
given by Inglis’ solution:
23
Griffith’s Energy Balance

Total energy:
Differentiating:
Equating and solving for critical stress 𝜎c :
Irwin defined the energy release rate as the
rate of change in potential energy due to
the crack area, given by:
24
Crack propagation occurs when 𝐺≥𝐺𝑐
Irwin showed a relationship between the
energy release rate and stress variation, given
by the stress intensity factor 𝐾
The energy release rate is compared to the fracture
energy 𝑤𝑓 required to generate two new surfaces,
and a critical value of energy release rate becomes:
Griffith’s Energy Balance and Irwin’s Strain Energy Release Rate

Conclusion (3) of Griffith’s paper indicates:
“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.”
i.e.,
σc √a = constant
The constant was independent of the size of the
crack (confirmed by Griffith experimentally)
The value of the constant was shown to be:
i.e.,
25
A.A. Griffith, “The Phenomena of Rupture and Flow in Solids”,
Philosophical Transactions of the Royal Society of London,
A 221, 163- 198 (1921) [ Presented February 26 1920].
A through-thickness crack in an infinitely wide plate
subjected to a remote tensile stress.

For a crack of length 2a in an infinite plate subject to a
remote tensile stress the energy release rate is given by:
G = πσ2 a/ E
where E is Young’s modulus, σ is the remotely applied
stress, and a is the half-crack length.
For the plate illustrated , the stress-intensity factor is
given by:
K I = σ √π a
(Fracture occurs when KI = Kic )
Combining:
G = K2
I /E
The energy and stress-intensity approaches to fracture
mechanics are essentially equivalent for linear elastic
materials.
• Irwin used the Westergaard approach to show that the
stresses and displacements near the crack-tip could be
described by a single constant that was related to the
energy release rate.
• This crack-tip characterizing parameter later became
known as the ‘‘stress-intensity factor”, K I , and
completely characterizes the crack-tip conditions in a
linear elastic material
• Fracture occurs at a critical stress
intensity Kic.
Kic is a measure of fracture toughness.
26
Through-thickness crack in an infinite plate subject to a remote tensile stress.
In practical terms, ‘‘infinite” means that the width of the plate is >> 2a.
Irwin, G.R., ‘‘Onset of Fast Crack Propagation in High Strength
Steel and Aluminum Alloys.” Sagamore Research Conference
Proceedings, Vol. 2, 1956, pp. 289–305.

Brittle Fracture: Elastic-Plastic Fracture:
27
Orowan - Irwin Modification of Griffith to Account for Plasticity

28
Fracture Toughness and Failure Regimes

Crack extension occurs when :
but crack growth may be stable or unstable,
depending on how G and wf vary with crack
size.
To illustrate stable and unstable behavior, it is
convenient to replace 2wf with R, the
material’s resistance to crack extension.
A plot of R vs. crack extension is called a
resistance curve or R curve.
The corresponding plot of G vs. crack
extension is the driving force curve.
The first R-curve 𝑅1 is a flat curve with a
critical energy release value 𝐺𝑐1
If the driving force curve exceeds this value
the crack become unstable, i.e,. 𝜎1 is stable,
no crack propagation occurs ; 𝜎2 is unstable
and crack propagation occurs.
The second R-curve 𝑅2 is a rising curve,
where 𝜎3 is stable and 𝜎4 is unstable; contrary
to the constant curve, the crack is allowed to
propagate to a critical value 𝑎𝑐2 with a
corresponding critical energy release rate 𝐺𝑐2. 29
Crack Resistance Curve (R curve) and
Crack Driving Force Curve (G Curve).
R1 is for a brittle material
R2 is for a ductile material

So far, two parameters that describe
the behavior of cracks: K and G.
K: local behavior (tip stresses)
G: global behavior (energy)
Irwin: for linear elastic materials, these
two parameters are uniquely related.
Crack closure analysis: work to open
the crack = work to close the crack
30
K-G Relationship

From Griffith’s theory of energy
balance, Irwin showed a relationship
between the energy release rate and
stress variation, given by the stress
intensity factor K.
The stress intensity factor is usually
given a subscript to denote the mode
of loading, i.e., KI, KII, or KIII.
Mode I: Load is applied normal to the
crack plane called opening mode.
Mode II: In-plane shear load, tends to
slide one crack face with respect to
the other.
Mode III: Out-of-plane mode, shear
load (anti-plane Strain)
31
Stress Intensity Factor and Crack Opening Modes

Two stress components are assumed:
A singular and a non-singular stress
component.
For the singular component, the stresses
are in the vicinity of the crack-tip;
For the non-singular component, the
stresses are away from the crack-tip .
Griffith found a relationship between the
variation of the singular stresses and the
distance from the crack-tip.
For a 1/√𝑟 singularity, when 𝑟→0 , the
stress approaches infinity.
Irwin developed the stress intensity
factor 𝐾 that describe the crack-tip
conditions relative to stress, strain and
displacement near the crack tip.
The stress intensity factor for each
loading mode with subscript I-III are
presented in the next slide.
32
Crack Tip Singularities
Stress variation in the vicinity of a crack tip

Principal stresses are those
stresses that act on principal
surface.
Principal surface here means
the surface where components
of shear-stress are zero.
Principal direction:
Principal stresses:
33
Principal Stresses

The singular stress fields are given as functions
of the three modes of loading from the polar
coordinate system 𝑟,𝜃
34
Crack Tip Stress Fields
Stress components in polar coordinates

35
Crack plane is a principal plane with the following
principal stresses:
On the crack plane:
Stresses on the Crack Plane

• K uniquely defines the crack
tip stress field
• Modes I, II and III:KI,KII,KIII
36
Similitude
Stress Intensity Factor (SIF)

For the stress intensity factor to be useful, K
must be determined from remote loads and
the geometry.
Closed-form solutions for K have been
derived for a number of simple configurations.
The general form is:
Where:
37
Global Stress Fields
Stress normal to the crack plane in Mode I.

Stress intensity factors for Mode I:
• For a through crack in an infinite
plate:
In general:
where:
38
Stress Intensity Factor for Mode I
Through Crack in an Infinite Plate

For a single edge crack specimen subject to Mode I and
Mode II loading, where the load is applied normal to the
crack front and as a shear load:
The relationships between crack length 𝑎and width of
specimen 𝑊, contribute significantly to the value of the
stress intensity factor, especially for Mode I
i.e.,
39
K for a Single Edge Cracked Specimen
Mode I and Mode II Loading

KI solutions are shown for several test specimens
(Table and Graph)
Several handbooks devoted solely to stress intensity
solutions have been published
40
KI Solutions for Common Test Specimens

41
KI Solutions for Common Crack Geometries

Irwin accounted for the softer material in
the plastic zone by defining an effective
crack length that is slightly longer than the
actual crack size:
ry for plane stress:
For plane strain, yielding is suppressed by
the triaxial stress state, and the Irwin plastic
zone correction is smaller by a factor of 3:
42
Crack Tip Plasticity - Irwin
The effective Mode I stress intensity factor for a through
crack in an infinite plate in plane stress is given by:-

43
Infinite plate with though thickness
crack 2a
• Plane stress condition
• Elastic perfectly plastic material
Hypotheses:
• All plastic deformation concentrates
in a line in front of the crack.
• The crack has an effective length
which exceeds that of the physical
crack by the length of the plastic
zone.
• ρ: chosen such that stress
singularity at the tip disappears.

Dugdale and Barenblatt obtained
the Strip-Yield model shown.
Shown is a plastic zone ,length 2𝜌,
for a crack in compression under a
load corresponding to the yield
stress.
From this a stress intensity factor
for closure stress is obtained, and
the length of the plastic zone is
close to that determined by Irwin:
Burdekin and Stone found that an
effective crack length 𝑎𝑒ff in the area
between 𝑎−(𝑎+𝜌) and determined
an expression for the effective
stress intensity factor.
A closed form solution for the
through crack is given by:
44
Strip-yield model with compressive
yield stress at the plastic zone

Variation of 𝐾𝑒ff of Irwin and the strip-yield
correction models (Dugdale, Burdekin) is
compared with the solution for a through crack
from LEFM in with respect to the normalized
stress and non-dimensionalized 𝐾𝑒ff
The 𝐾𝑒ff of Irwin and the strip-yield model
deviate from the LEFM solution at stresses
greater than 0.5 𝜎𝑦ield
The two corrected models agree with each
other up to a value of approximately 0.85 𝜎𝑦ield
beyond which, the strip yield model
approaches infinity.
The LEFM approach is ensured if the plastic
zone is small compared to the dimensions of
the geometry.
Restriction due to crack size and stress
variation have been suggested in order to
ensure the LEFM approach:
46
Strip Yielding - Limits

47
Typical stretch zone in (a) untilted and (b) 45o tilted view
and (c) & (d) the measurement procedure with reference
locations
Physical Metallurgy of Crack Tip Plasticity
SEM Measurements of SZW
Geometrical inter-relation between
normal and tilted configuration of specimen

In Plane Stress, the axiallity of the
stress state decreases due to a
reduction in 𝜎𝑧𝑧 near the surface.
From these stress variations at a
crack-tip, and by setting 𝜃=0, results in
the following for x, y and z stress
components:
For Plane Stress:
For Plane Strain:
48
3D edge crack with interior and
surface coordinate system,s
Stress variation along the z-axes
of the 3D edge crack

• Plane stress failure: in general,
ductile
• Plane strain failure: in general,
brittle
49

50

51
Changes in Crack Direction
F. Erdogan and G. C. Sih, “On the Crack Extension in Plates
Under Plane Loading and Transverse” ASME, 1963.

Plane Stress Plane Strain
52
DHC in Zr-2.5wt.% Nb Compact Tension Specimens;
Timmins, P., “Effect of Stress State and Texture on DHC Kinetics”,
ECF 9, Varna, Bulgaria, 21-25 September 1992
In Plane Stress, the crack tunnels along the narrow region of Plane Strain at the centre
of the specimen. The crack front propagated an order of magnitude faster in Plane Strain

Hahn and Rosenfield’s
Conclusions from their 1964 paper
are as follows:
“1. Yielding is predominantly of the
plane strain plastic-hinge type until
the extent of the yielded zone is
about equal to the sheet thickness.
Further deformation, under plane
stress conditions, proceeds by a
45-degree–shear mode.
2. The general shape of the 45-
degree shear zone can approach
that of the DM (Dugdale-
Muskhelishvili) crack model.
Predictions of this model are in
agreement with measured zone
size and displacement values for
silicon steel.”
(How cool is this? P.T.)
53
Plastic Zones – G.T Hahn and A.R. Rosenfield, “Local Yielding and
Extension of a Crack Under Plane Stress”, Washington D.C, 1964

54
Hall-Petch Relationship for Stress Intensity Factor, K
R.W. Armstrong, in: Eng. Fract. Mech., 28, [5-6], 529 (1987)

Linear Elastic Fracture Mechanics -Examples

57
Cylindrical Pressure Vessel with an
Inclined Through-thickness Crack

58
This is why an overcooked Chayvo hotdog
usually cracks along the longitudinal direction
first (i.e. its skin fails from hoop stress, generated
by internal steam pressure).
Cylindrical Pressure Vessel with an
Inclined Through-thickness Crack

60
Worked Example -1 - Solution

62
Consider an infinite plate with a central crack of length 2a subjected to a
uniaxial stress perpendicular to the crack plane. Using Irwin’s model for a
plane stress case, show that the effective SIF is given as follows:
Solution:
The effective crack length is:
The effective SIF is thus:
With:
Worked Example -3

Elastic-Plastic Fracture Mechanics
(EPFM)

64
EPFM Regime – Crack Tip Plasticity

The energy release rate is the rate of change in
potential energy due to the crack area:
Or, more notably:
Where μ = shear modulus and Jc = 2ϒ where ϒ =
surface tension.
The J-integral is cable of describing the elastic-
plastic (EPFM) behavior which is not the case for
energy release rate and can be described by a path
around a crack tip , as follows:
Where:
65
Contour Integral
Rice, J.R., “Some Remarks on Elastic Crack-Tip Stress Fields.” International Journal of Solids and
Structures, Vol. 8, 1972, pp. 751–758.
G.P. Cherepanov, “Mechanics of Brittle Fracture,” McGraw-Hill, New York (1979).

Determination of the energy release rate by J-integral is
performed for a double cantilever beam specimen, where
the dotted line from B to G is the contour Γ of the J-
integral :
Consider the following approximations:
• The strain energy of the system is negligible.
• CD, DE and EF are free surfaces, so no traction vector
• P is replaced by a shear load, to become a traction
load at BC and FG.
• No displacement or traction in 𝑥direction.
Displacement and traction in 𝑦 direction for BC and FG,
the J-integral is given by:
Where:
And:
66
Example - Energy Release Rate by J-integral for a Double
Cantilever Beam Specimen
The traction in 𝑦𝑦direction for 𝐵C and 𝐹G is given by
𝑇𝑦=𝑃/𝐵 and the J-integral:
With E =210 GPa:

CTOD can be defined as the crack-opening
displacement at the end of the strip-yield zone.
According to this definition, CTOD
in a through crack in an infinite plate subject to
remote tensile stress is given by:
As:
67
Estimation of CTOD from the Strip Yield Model.
Burdekin, F.M. and Stone, D.E.W., “The Crack Opening Displacement Approach to Fracture
Mechanics in Yielding Materials.” Journal of Strain Analysis,Vol. 1, 1966, pp. 145–153.

Since J = G for linear elastic material
behavior, then the relationship
between CTOD and J in the limit of
small-scale yielding is given by:
where m is a dimensionless constant
that depends on the stress state and
material properties.
It can be shown that the preceding
relationship applies well beyond the
validity limits of LEFM.
Since the strip-yield model assumes
σy = σYS within the plastic zone, the
J-CTOD relationship is given by:
68
J-CTOD Relationships

One measure of fracture toughness JIc is defined
near the initiation of stable crack growth.
The precise point at which crack growth begins
is usually ill-defined. Consequently, the definition
of
JIc is somewhat arbitrary, much like a 0.2%
offset yield strength.
The corresponding CTOD near
the initiation of stable crack growth is denoted δi
by U.S. and British testing standards.
69
Crack-Growth Resistance Curves

The ASTM procedure for computing JQ, a
provisional JIc, from the R curve is shown.
Exclusion lines are drawn at crack extension
(Δa) values of 0.15 and 1.5 mm. These lines
have a slope of MσY, where σY is the flow
stress, defined as the average of the yield and
tensile strengths.
The slope of the exclusion lines is intended to
represent the component of crack extension
that is due to crack blunting, as opposed to
ductile tearing. The value of M can be
determined experimentally, or a default value of
2 can be used. A horizontal exclusion line is
defined at a maximum value of J:
All data that fall within the exclusion limits are fit
to a power-law expression:
JQ = JIc as long as the following size
requirements are satisfied:
70
JIc Measurement

71
J1.5mm -Prediction from Microstructure:
Timmins, P. “Effect of Heat Treatment on the Ductile Fracture
Characteristics of Zr-2.5 wt.% Nb”, ECF 8, Turin, Italy, Oct 1-5, 1990
Arrows indicate circles containing about
3 major voids to give an average of about 12 microns

74
Specimens most commonly used in ductile fracture experiments, corresponding stress
triaxiality levels and measured fracture properties.
Smooth, Notched and Cracked Specimens
Used in Ductile Fracture Studies

75
Phenomenological Modeling of Ductile Fracture in Steel

76
SEM Fractographs after Image Processing

77
Spherical void in a solid, subject to triaxial
stress state
The limit load model for void instability. Failure is
assummed to occur when the net section stress
between voids reaches a critical value
DUCTILE FRACTURE: VOID NUCLEATION, GROWTH, and COALESCENCE
DUCTILE FRACTURE: VOID NUCLEATION, GROWTH, and COALESCENCE
Ductile Fracture: Void Nucleation, Growth, and Coalescence

78
Ductile Fracture in Round Notched Bars of High Strength Steel:
Damage Accumulation, Initiation of Macroscopic Crack, Crack Growth and
Shear Lip Formation.

In plane strain bars where slant fracture
prevails, the shear lips cover almost the
entire fracture surface
79
Slant Fracture

80
Ductile crack growth in a 45° zig-zag pattern Optical micrograph of ductile crack growth in a high
strength-low alloy steel
Zig-Zag Crack Growth by Void Nucleation, Growth, and Coalescence

81
More Ductile Crack Propagation by Void Growth and Coalescence
Round notched bar of low alloy steel:
The zig-zag crack goes “slant” as it approaches the free surfaces forming the shear lips
that characterize cup-cone fracture.

82
Modes of void coalescence. (a) Necking of intervoid ligament or coalescence in a
layer (b),(c) Coalescence in a micro-shear band (“void-sheet” coalescence)
(d)“Necklace” coalescence or coalescence in columns (voided columns).
Major loading axis is vertical in all. Loading is axisymmetric in (a)–(c) and plane strain in (d).
Modes of Void Coalescence – MnS in C-Mn Steels

83
Ductile growth of an edge crack. The shear lips are produced by the same mechanism as the cup and cone in
uniaxial tension.
The crack tunnels along the region of Plane Strain at the centre of the specimen.
Crack Growth by Void Nucleation, Growth, and Coalescence-
Tunneling

Plane Stress Plane Strain
84
DHC in Zr-2.5wt.% Nb Compact Tension Specimens;
Timmins, P., “Effect of Stress State and Texture on DHC Kinetics”,
ECF 9, Varna, Bulgaria, 21-25 September 1992 - Tunneling
In Plane Stress, the crack tunnels along the narrow region of Plane Strain at the centre
of the specimen. The crack front propagated an order of magnitude faster in Plane Strain

• Ductile crack tunneling and
transition from flat to slant
fracture in pre-cracked
specimens can take place when
the plastic zone size is greater
than the specimen thickness, as
is often the case in wide
panels/plates or thin sheets.
• In Plane Stress, the crack
tunnels along the narrow region
of Plane Strain at the centre of
the specimen.
85
Ductile Crack Tunneling and Transition from Flat to Slant
Fracture in Pre-cracked Specimens
Ductile growth of an edge crack. The shear lips are produced by
the same mechanism as the cup and cone in uniaxial tension

86
Crack Growth Ahead of a Crack Tip by Void Nucleation, Growth,
and Coalescence
Ductile growth of an edge crack.

88
Dislocation Models of Cleavage Fracture
Zener (1948)
Stroh (1957)
Cottrell (1958)

89
• Zener’s and Stroh’s models predict that crack formation is the most
difficult stage in fracture, so the fracture is initiation-controlled.
• This is at variance with the experimental results for most materials.
• Cottrells' model predicts a tensile stress controlled cleavage.
• It also explains the effect of grain size and yielding parameters on fracture.
• Cottrell’s model can explain cleavage fracture in single crystals, which does
not involve grain boundaries as barriers to dislocation pile-ups.
Comments on Dislocation Models of Cleavage Fracture

90
One model of cleavage fracture
in steels indicates initiation of
cleavage at a microcrack that
forms in a second phase
particle ahead of the
macroscopic crack.
Smith, E., “Physical Basis of
Yield and Fracture”, Conference
Proceedings, p. 36, Institute of
Physics and Physical Society,
London, 1966
Cleavage Initiation in Steels – Smith’s Cracked Carbide Model

91
Smith’s Cracked Carbide Model(1966)

92
Fracture Initiation Ahead of Notch Root (J. F. Knott et alia)
at Some “Characteristic Distance” at which
Cleavage Fracture is “Triggered”

Ritchie, R.O., Knott, J.F., and Rice,
J.R.,‘‘On the Relationship between
Critical Tensile Stress and Fracture
Toughness in Mild Steel.” Journal of
the Mechanics and Physics of
Solids, Vol. 21, 1973, pp. 395–410
The Ritchie-Knott-Rice model for
cleavage fracture. Failure is
assumed to occur when the fracture
stress is exceeded over a
characteristic distance.
93
Ritchie-Knott-Rice Model for Cleavage Fracture -
Characteristic Distance

• When a flawed structure is subject to an
applied K, a microcrack may or may not
initiate, depending on the temperature as
well as the location of the eligible
cleavage triggers.
• The initiation of cleavage cracks should
be governed by a weakest link
mechanism, because the process
involves searching for a large enough
trigger to propagate a microcrack into the
first ferrite grain.
• Once cleavage initiates, the crack may
either propagate in an unstable fashion or
arrest, as shown.
• Initiation is governed by the local stress
at the critical particle, while propagation is
controlled by the orientation of the
neighboring grains and the global driving
force.
• The overall probability of failure is
equal to the probability of initiation times
the conditional probability of propagation.
95
Examples of unsuccessful cleavage events: (a) arrest at particle/matrix interface, (b) arrest at
a grain boundary, and (c) arrest due to a steep stress gradient.
Weakest Link -2

• In specimens that exhibited low
toughness, this distance was small; a
critical nucleus was available near the
crack tip.
• In specimens that exhibited high
toughness, there were no critical
particles near the crack tip; the crack
had to grow and sample additional
material before a critical cleavage
nucleus was found.
• Shown is a plot of fracture toughness
vs. the critical distance rc, which
Heerens and Read measured from the
fracture surface; rc is defined as the
distance from the fatigue crack tip to the
cleavage initiation site.
• The resistance curve for ductile crack
growth is also shown in this plot.
• In every case, cleavage initiated near
the location of the maximum tensile
stress
96
Cleavage Trigger - Heerens, J. and Read, D.T., ‘‘Fracture Behavior of a Pressure Vessel Steel
in the Ductile-to-Brittle Transition Region.” NISTIR 88-3099, National Institute for Standards and
Technology, Boulder, CO, 1988.
Relationship between cleavage fracture toughness and the
distance between the crack tip and the cleavage trigger.

97
SEM fractograph of cleavage in an A 508 steel
SEM Fractograph of Cleavage in an A 508 Steel
Formation of river patterns, as a result of a
cleavage crack crossing a twist boundary
between grains. Note the tearing lines (light areas) between parallel cleavage planes

• The fracture toughness of ferritic steels can
change drastically over a small temperature
range, as shown.
• At low temperatures, steel is brittle and fails by
cleavage.
• At high temperatures, the material is ductile and
fails by microvoid coalescence. Ductile fracture
initiates at a particular toughness value, as
indicated by the dashed line.
• The crack grows as the load is increased.
Eventually, the specimen fails by plastic collapse
or tearing instability.
• In the transition region between ductile and
brittle behavior, both micromechanisms of fracture
can occur in the same specimen.
• In the lower transition region, the fracture
mechanism is pure cleavage, but the toughness
increases rapidly with temperature as cleavage
becomes more difficult.
• In the upper transition region, a crack initiates
by microvoid coalescence but ultimate failure
occurs by cleavage.
• On initial loading in the upper transition region,
cleavage does not occur because there are no
critical particles near the crack tip. As the crack
grows by ductile tearing, however, more material
is sampled. Eventually, the growing crack samples
a critical particle and cleavage occurs.
• Because the fracture toughness in the transition
region is governed by these statistical sampling
effects, the data tend to be highly scattered.
99

100
Ductility Transition from Void Coalescence to
Cottrell Cracked Dislocation
J.P. Gudas, G.R. Irwin, R.W. Armstrong and X.J. Zhang, in: Defect Assessment in components – fundamentals and applications,
eds. J.G. Blauel and K.-H Schwalbe ESIS/EGF9 (Mech. Eng. Publ. Ltd, London, 1991) pp. 549-568
7

• Cleavage propagation in the upper transition
region often displays isolated islands of ductile
facture.
• When specimens with arrested macroscopic
cleavage cracks are studied metallographically,
unbroken ligaments are sometimes discovered
behind the arrested crack tip.
• These two observations imply that a
propagating cleavage crack in the upper
transition region encounters barriers, such as
highly misoriented grains or particles, through
which the crack cannot propagate.
• The crack is diverted around these obstacles,
leaving isolated unbroken ligaments in its
wake.
• As the crack propagation continues, and the
crack faces open, the ligaments that are well-
behind the crack tip ruptures
101
Cleavage crack propagation in the ductile-brittle transition region.
Ductile ligaments rupture behind the crack tip, resulting in
increased propagation resistance.
Cleavage Crack Propagation in the Ductile-Brittle Transition Region.

102
Correlation of Tensile and Charpy Ductile-Brittle Transition Results
R.W. Armstrong, G.R. Irwin and X.J. Zhang, in Cleavage Fracture; George R. Irwin Symposium
(TMS-AIME, Warrendale, PA, 1998) pp. 51-58

The first rigorous mechanics-based
fatigue crack nucleation criterion
proposed is likely that developed by
Stroh in 1954. In this model, shown
schematically opposite, a line of discrete
dislocations, forming a persistent slip
band (PSB), is contained within an
infinite elastic medium and inclined at a
given angle to a remote nominal stress,
σ0 . The resulting normal stresses σn at
the termination of the PSB are
developed in terms of the length of the
PSB, l, its orientation,ϴ, and the
distance from its end, r, as follows:
Where: ϴ= 70.5 degrees
103
Schematic diagram of the Stroh crack nucleation model
showing a PSB of length l, orientated to a remote stress
and in an infinite elastic medium
Stroh, AN. The formation of cracks as a result of plastic flow.
Proc. Roy. Soc. Lond., 223, 404-414, 1954.

104
Changes in Crack Direction
F. Erdogan and G. C. Sih, “On the Crack Extension in Plates
Under Plane Loading and Transverse” ASME, 1963.
The maximum Tangential Shear Stress is at 70.5 degrees, so that as a crack propagates,
the cleavage crack direction changes as the contribution of these components changes from
grain to grain. In Stroh’s model, Crack Nucleation is the limiting event.

106
There is no single mechanism for intergranular fracture. Rather, there are a
variety of situations that can lead to cracking on grain boundaries, including:
1. Precipitation of a brittle phase on the grain boundary
2. Hydrogen embrittlement and liquid metal embrittlement
3. Enviromental assisted cracking
4. Intergranular corrosion
5. Grain boundary cavitation and cracking at high temperatures
Ductile metals usually fail by coalescence of voids formed at inclusions and
second phase particles
Brittle metals typically fail by transgranular cleavage
Under special circumstances, HOWEVER, cracks can form and propagate along
grain boundaries resulting in intergranular fracture

107
Intergranular fracture in a steel ammonia tank
Brittle phases can be deposited on
grain boundaries of steel as a result
of improper tempering: tempered
martensite embrittlement (tempering
at 350 °C). Involves segregation of
impurities (P, S) to prior austenite
grain boundaries.
Atomic hydrogen bonds with the
metal atoms reducing the cohesive
energy strength at grain boundaries.
Sources: H2S, hydrogen gas.
Important problem in welding of
steels: cracking in the Heat Affected
Zone (HAZ). Hydrogen is a problem
when welding high strength steels.
Intergranular Fracture - Examples

109
Laird (1967) model of plastic
blunting-re-sharpening wich
leads to fatigue crack growth in
reverse plane bendingfatigue.
a: zero load
b: small tensile load
c: peak tensile load
d: onset of load reversal
e: peak compressive load
f: smal tensile load in the
subsequent tensile cycle.
Arrows indicate slip direction Fatigue Striations
of Failure Surface
in 2024-T3
Aluminium alloy.
Arrow indicates
growth direction
Mechanism of Fatigue Crack Growth

110
5 mm
Fatigue striations
2 mm
Beach marking on a fatigue fracture surface in a thin walled pipe
Fatigue Crack Growth- Beach Marks

111
Fracture surface of high strength
Al 2024 - T3 specimen which failed by
fatigue.
Test specimen was a Centre Notch-
panel 610 mm x 229 mm, 10 mm
thickness with initial crack length
13 mm.
Arrow indicates direction of crack
growth.
Image corresponds to a position 20
mm from the center of the plate.
EXAMPLE: Striation Width vs. da/dN

112
Block loading sequence
Block A
13 MPa m1/2
Block A: 0.5 mm / cycle
Block B: 0.34 mm / cycle
Block C: 0.05 mm / cycle
(Da / DN)mean
Block A, R = 0.5: DKeff = 0.75 DK
DK = 17 MPa m1/2, Ds?, smax?, smin?

Mechanisms of Stress Corrosion Cracking
and Hydrogen Embrittlement

In order for the crack to propagate
by this mechanism, the corrosion
rate at the crack tip must be much
greater than the corrosion rate at
the walls of the crack.
If the crack faces and crack tip
corrode at similar rates, the crack
will blunt. Under conditions that are
favorable to SCC, a passive film
(usually an oxide) forms on the
crack walls.
This protective layer suppresses
the corrosion reaction on the crack
faces.
High stresses at the crack tip
cause the protective film to rupture
locally, which exposes the metal
surface to the electrolyte, resulting
in crack propagation due to anodic
dissolution
115
Simple illustration of anodic SCC.
The crack-tip corrosion rate must be much greater than the
corrosion rate at the crack walls.
Such a condition requires that a passive film form on the crack
walls.
Anodic Stress Corrosion Cracking- Mechanism

Hydrogen embrittlement is responsible for much
of what has traditionally been referred to as
“stress corrosion cracking.” For example,
environmental cracking of high strength steel,
aluminum,
and titanium alloys in aqueous solutions is
usually driven by hydrogen production at the
crack tip
(i.e., the cathodic reaction) rather than anodic
SCC.
116
Cathodic Stress Corrosion Cracking- Hydrogen Embrittlement
Mechanism
Hydrogen is concentrated at the fracture process zone near the crack tip.
The high degree of stress triaxiality near the crack tip causes the crystal lattice to expand, which increases
the hydrogen solubility locally.
The high local concentration of hydrogen causes the process zone to be embrittled. This embrittlement,
along with the high local stresses, results in microcracking in the process zone. The microcracks that form
in the process zone link up with the main crack, resulting in crack extension. The main crack propagates
over time, as the local crack-tip processes of hydrogen uptake and microcracking occur continuously

118
The first CTOD test standard was published in Great Britain in 1979. Several years later, ASTM
published E 1290, an American version of the CTOD standard.
ASTM E 1290 has been revised several times, and the most recent version (as of this writing)
was published in 2002.
The original British CTOD test standard has been superceded by BS 7448, which combines K, J,
and CTOD testing into a single standard.
ASTM E 1820 also combined these three crack tip parameters into a single testing standard, but
E 1290 is still maintained by the ASTM Committee E08 on Fatigue and Fracture.
The CTOD test methods in E 1290 and E 1820 are similar, but the latter standard includes
provisions for generating a CTOD resistance curve.
ASTM E 1820 includes both a basic and resistance curve procedure for CTOD, much like the J
test methodology in this standard. The test method in E 1290 is comparable to the basic
procedure.
Experimental CTOD estimates are made by separating the CTOD into elastic and plastic
components, similar to J tests.
Overview and Evolution of Testing

Testing of the specimen is conducted by
applying a load 𝑃; the load is increased with a
speed that ensures a quasi-static conditions
until the specimen reaches fracture.
During the test, measurements of the load 𝑃
and the crack mouth opening displacement
(CMOD) Δ are conducted.
To ensure the correct value of the load to
determine the fracture toughness, a
corresponding load 𝑃𝑄 is used.
If:
Then
119
Fracture Toughness, 𝐾𝐼c , Testing for Mode I - ASTM E399

ASTM procedure for computing JQ, a
provisional JIc, from the R curve is
illustrated.
Exclusion lines are drawn at crack
extension (Δa) values of 0.15 and 1.5
mm.
These lines have a slope of MσY,
where σY is the flow stress, defined
as the average of the yield and
tensile strengths. The slope of the
exclusion lines is intended to
represent the component of crack
extension that is due to crack
blunting, as opposed to ductile
tearing. The value of M can be
determined experimentally, or a
default value of 2 can be used.
If:
Then:
120
Determination of JQ from a J-R curve. Taken from E 1820-01,
‘‘Standard Test Method for Measurement of Fracture Toughness.’’
American Society for Testing and Materials, Philadelphia, PA, 2001.
J Testing : BS 7448 Part 1 and ASTM E 1820.

Experimental CTOD estimates are
made by separating the CTOD into
elastic and plastic components, similar
to J tests. The elastic CTOD is
obtained from the elastic K:
The plastic displacement at the crack
mouth, Vp, is related to the plastic
CTOD through a similar triangles
construction:
For the ith crack size:
121
CTOD Testing
Hinge model for plastic displacements in an SE(B) specimen.

122
Weldments have highly heterogeneous microstructures
Fracture toughness can vary considerably over relatively short distances. Thus, it is important to take great care in
locating the fatigue crack in the correct region. If the fracture toughness test is designed to simulate an actual
structural flaw, the fatigue crack must sample the same microstructure as the flaw. For a weld procedure
qualification or a general assessment of a weldment’s fracture toughness, location of the crack in the most brittle
region may be desirable, but it is difficult to know in advance which region of the weld has the lowest toughness.
In typical C–Mn structural steels, low toughness is usually associated with the coarse-grained heat-affected zone
(HAZ) and the intercritically reheated HAZ. A microhardness survey can help identify low toughness regions
because high hardness is often coincident with brittle behavior. The safest approach is to perform fracture
toughness tests on a variety of regions in a weldment.
For weld metal testing, the through-thickness orientation is usually preferable because a variety of regions in the
weld are sampled. However, there may be cases where the surface-notched specimen is the most suitable for
testing the weld metal. For example, a surface notch can sample a particular region of the weld metal, such as the
root or cap, or the notch can be located in a particular microstructure, such as unrefined weld metal.
Notch location in the HAZ often depends on the type of weldment. If welds are produced solely
for mechanical testing, for example, as part of a weld procedure qualification or a research program, the welded
joint can be designed to facilitate HAZ testing
Fracture Testing of Weldments - Overview

123
Fracture Testing of Weldments – Notches in Specimens
Notch orientation in weldment specimens. (a) through-thickness notch and (b) surface notch.
Taken from Dawes, M.G., Pisarski, H.G., and Squirrell, H.G., ‘‘Fracture Mechanics Tests on Welded Joints.’’
ASTM STP 995, American Society for Testing and Materials, Philadelphia, PA, 1989, pp. II-191–II-213.

124
Fracture Testing of Weldments – Post-Test Microstructures
Posttest sectioning of a weldment fracture toughness specimen to identify the microstructure
that caused fracture.

ASTM Standard E 23 covers
Charpy and Izod testing. These
tests both involve impacting a
small notched bar with a pendulum
and measuring the fracture energy.
The Charpy specimen is a simple
notched beam that is impacted in
three-point bending, while the Izod
specimen is a cantilever beam that
is fixed at one end and impacted at
the other.
A number of investigators have
attempted to correlate Charpy
energy to fracture toughness
parameters such as KIc. Some of
these empirical correlations seem
to work reasonably well, but most
correlations are often unreliable.
125
Charpy and Izod notched impact tests. Taken from E 23-02a,
‘‘Standard Test Methods for Notched Bar Impact Testing of Metallic Materials”.
American Society for Testing and Materials, Philadelphia, PA, 2002.
Charpy and Izod Impact Testing - ASTM Standard E 23

For constant amplitude loading, the crack
growth rate can be described by:
Crack growth rate behaviour for constant
amplitude loading is described by a
sigmoidal curve opposite, where a
log-log plot of da⁄dN and Δ𝐾 is shown.
127

The sigmoidal curve contains three distinct regions. At intermediate ΔK
values, the curve is linear, but the crack growth rate deviates from the
linear trend at high and low ΔK levels. At the low end, da/dN approaches
zero at a threshold ΔK, below which the crack will not grow.
In some materials, the observed growth rate increases rapidly at high ΔK
values. There are two possible explanations for the Region III behavior.
Some researchers have hypothesized that the crack growth rate
accelerates as Kmax approaches Kic, the fracture toughness of the
material. According to this hypothesis, microscopic fracture events (e.g.,
pop-ins) contribute to crack growth, resulting in a higher overall growth
rate.
An alternative hypothesis is that the apparent acceleration in da/dN is not
real but is due to the influence of crack-tip plasticity on the true driving
force for fatigue. At high Kmax values, linear elastic fracture mechanics
is no longer valid, and a parameter like ΔJ might be more appropriate to
characterize fatigue.
The linear region of the log-log plot in can be described by a power law:
where C and m are material constants that are determined
experimentally. Accordingly, the fatigue crack growth rate depends only
on ΔK; da/dN is insensitive to the R ratio in Region II.
128

Region 1:
Klesnil and Lukas together with NASGRO
cover this region:
Region 2:
Paris, Forman and NASGRO cover this
region and behave uniformly:
Region 3
Forman and NASGRO cover this region, they
obtain a value that goes to infinity at the
fracture toughness value
129
Fatigue Crack Growth Models

Fatigue crack growth data near the threshold
for mild steel at various R ratios, is shown.
(From:
Tanaka, K., “Mechanics and Micromechanics of
Fatigue Crack Propagation.” ASTM STP 1020,
American Society for Testing and Materials,
Philadelphia, PA, 1989, pp. 151–183.)
When data at lower R ratios are corrected for
closure, the R ratio effect disappears and all
data exhibit the same threshold, which
corresponds to ΔKth for the material.
This effect presents strong evidence in favor of
the closure mechanism in the threshold range.
Crack closure is thought to decrease the fatigue
crack growth rate by reducing the effective
stress-intensity range.
130
Near Threshold Data for C-Mn Steel

In Region II, where da/dN follows
a power law, the crack growth
rate is relatively insensitive to
microstructure and tensile
properties, while da/dN at either
extreme of the curve is highly
sensitive to these variables.
One explanation for the lack of
sensitivity to metallurgical
variables is that cyclic flow
properties, rather than monotonic
tensile properties, control fatigue
crack propagation.
131
Summary of Fatigue Crack Growth Regions

132
Figure (a) illustrates the procedure for determining the first inspection interval in the
structure.
The lower curve defines the “true” behavior of the worst flaw in the structure, while
the predicted curve assumes the initial flaw size is ao.
The time required to grow the flaw from ao to at (the tolerable flaw size) is computed. The first
inspection interval I1 should be less than this time, in order to preclude flaw growth beyond at
before the next inspection.
If no flaws greater than ao are detected, the second inspection interval I2 is equal to I1, as Figure
(b) illustrates.
Suppose that the next inspection reveals a flaw of length a1, which is larger than ao.
In this instance, a flaw growth analysis must be performed to estimate the time required to grow
from a1 to at.
The next inspection interval I3 might be shorter than I2, as Figure (c) illustrates.
Inspection intervals would then become progressively shorter as the structure approaches the
end of its life.
The structure is repaired or taken out of service when the flaw size reaches the maximum
tolerable size, or when required inspections become too frequent to justify continued operation.
Defect Tolerance Analysis - (Time Dependent Flaw Growth)

133
Schematic damage tolerance
analysis:
(a) determination of first inspection
interval I1,
(b) determination of second
inspection interval I2, and
(c) determination of third inspection
interval I3.

Introduction to Failure Assessment Diagrams

API 579 Fitness For Service Using Failure Assessment Diagrams
135

Idealized Failure Assessment Diagram (FAD)
Failure Assessment Diagram (FAD)
136

Data - KIC value of 56 MPa√m
Tendency is towards LEFM (Linear Elastic Fracture Mechanics) behavior –i.e., elastic (brittle)
fracture and/or EPFM (Elastic-Plastic Fracture Mechanics) behavior-i.e., mixed elastic (brittle)
fracture and plastic deformation.
Failure Assessment Diagram (FAD) – Effect of Low KIC value
Example: TEG Contactor Inlet Filter Separators
137

Data - KIC value of 120 MPa√m
Tendency is towards Plastic Collapse – i.e., separation by plastic deformation (no brittle behavior)
Failure Assessment Diagram (FAD) – Effect of High KIC value
Example: TEG Contactor Inlet Filter Separators
138

The curve represents the locus of predicted failure
points.
Fracture is predicted when Keff = Kmat, where
Kmat is the fracture toughness in terms of stress
intensity units. If the toughness is very large, the
structure fails by collapse when Sr = 1.0.
A brittle material will fail when Kr = 1.0. In
intermediate cases, collapse and fracture interact,
and both Kr and Sr are less than 1.0 at failure. All
points inside of the FAD are considered safe; points
outside of the diagram are unsafe.
In order to assess the significance of a particular
flaw in a structure, one must determine the
toughness ratio as follows:
The stress ratio for the component of interest can
be defined as the ratio of the applied stress to the
collapse stress. Alternatively, the applied Sr can be
defined in terms of axial forces or moments.
If the assessment point with coordinates (Sr , Kr)
falls inside of the FAD curve, the analysis predicts
that the component is safe.
139
The strip-yield failure assessment diagram. Taken from Dowling, A.R. and
Townley, C.H.A., “The Effects of Defects on Structural Failure: A Two-Criteria
Approach.” International Journal of Pressure Vessels and Piping, Vol. 3,
1975, pp. 77–137; Harrison, R.P., Loosemore, K., and Milne, I., “Assessment
of the Integrity of Structures Containing Defects.” CEGB Report R/H/R6,
Central Electricity Generating Board, UK, 1976.
The Strip-Yield Failure Assessment Diagram - CEGB

A weldment is said to be overmatched when the
weld metal has higher strength than the base
metal. The reverse situation is known as an
undermatched weldment.
The mismatch in strength properties affects the
crack driving force in the elastic-plastic and fully
plastic regimes. Mismatch in properties is
normally not a significant issue in the elastic
range because the weld metal and base metal
typically have similar elastic constants.
Opposite is a schematic plot of the crack driving
force for a crack in a base metal as well as for a
crack of the same size in an overmatched weld.
Because the weld metal has higher yield
strength than the base metal, the upswing in the
driving force curve occurs at a higher load in the
weldment.
At a fixed load in the elastic-plastic regime, the
driving force in the cracked weldment is
significantly lower than in the cracked base
plate.
The effect of weld strength mismatch can be
taken into account in the FAD method through
an appropriate definition of Lr, as shown.
The reference stress for a weldment should be
defined from the elastic-plastic J solution.
140
Welded Structures – Weld Mismatch FAD Example
Effect of weld strength mismatch on crack driving force.
Strength mismatch effects can be taken into account in the FAD method
through the reference stress solution for the weldment. In this schematic,
Weld residual stress is neglected, and the weld and base metal are
assumed to have similar hardening characteristics

References
PROGRESS OF HIGH PERFORMANCE STEEL PLATES -
Nippon Steel & Sumitomo Metal Technical Report no. 110 September 2015
EFFECT OF HEAT TREATMENT ON THE EMBRITTLEMENT OF DISSIMILAR WELDED JOINTS,
Michael Francis Dodge, Ph.D. Thesis, University of Leicester , 2014
RUSSIAN METALS FOR ARCTIC OFFSHORE STRUCTURES,
Pavel Layus , M.Sc. Thesis, Lappeenranta University of Technology , 2012
MECHANISMS OF INCLUSION EVOLUTION AND INTRA-GRANULAR ACICULAR FERRITE
FORMATION IN STEELS CONTAINING RARE EARTH ELEMENTS,
Xiaoxuan Deng, Acta Metall. Sin.(Engl. Lett.)Vol.25 No.3 pp241-248 June 2012
THE EFFECT OF WELDING SPEED ON THE PROPERTIES OF
ASME SA516 GRADE 70 STEEL,
Alicia Hall, M.Sc. Thesis, University of Saskatchewan, 2010
CONSEQUENCES OF PWHT REQUIREMENTS ON SERVICE PROPERTIES FOR PETROCHEMICAL
GRADES,
Cédric CHAUVY, Lionel COUDREUSE and Philippe BOURGES - ARCELORMITTAL INDUSTEEL, CRMC,
Le Creusot, France, 2008
IMPROVEMENT OF WELD HAZ TOUGHNESS AT LOW HEAT INPUT BY CONTROLLING THE
DISTRIBUTION OF M-A CONSTITUENTS,
Risto Laitinen, Ph.D. Thesis, University of Oulu, 2006 141

ACICULAR FERRITE AND BAINITE MICROSTRUCTURE PROPERTIES AND COMPARISON OF THEIR
PHYSICAL METALLURGY RESPONSE
Eva Mazancová, Metal 2005
EFFECT OF ACICULAR FERRITE PRODUCED BY HEAT TREATMENT ON TOUGHNESS OF API 5L
X65 STEEL PIPE
Faris Naufal, Dick F. Firdaus, Nuke F. Prasiwi, Bondan T. Sofyan, Myrna A. Mochtar , International Journal
of Advances in Science and Technology (IJAST), 2005
HIGH-STRENGTH LINEPIPES WITH EXCELLENT HAZ TOUGHNESS
Yoshio TERADA, Akihiko KOJIMA, Akihito KIYOSE, Takao NAKASHIMA, Naoki DOI, akuya HARA, Hiroshi
MORIMOTO, Masaaki SUGIYAMA,
NIPPON STEEL TECHNICAL REPORT No. 90 JULY 2004
WELDING METALLURGY, Second Edition. Sindo Kou, John Wiley & Sons, Inc., 2003
ACICULAR FERRITE FORMATION IN A MEDIUM CARBON STEEL WITH A TWO STAGE CONTINUOUS
COOLING,
I Madariaga, CENIM, Madrid, Spain, 1999.
MODELING THE EVOLUTION OF MICROSTRUCTURE IN STEEL WELD METAL
H. K. D. H. Bhadeshia, University of Cambridge, Materials Science and Metallurgy, 1993
References
142

International FAD-Based Procedures for Fitness For Service
Assessments of Structures Containing Crack-Like Flaws

144
British Standards Institute (BSI) has published BS 7910:1999 and the European
Union conducted a cooperative research program that culminated in the publication
of the SINTAP document, which is an abbreviation for “structural integrity
assessment procedures for European industry.”
The CEGB R6, BS 7910, and SINTAP methods are very similar to one another,
probably because many of the same individuals were involved in creating all
the three documents.
In the U.S., the American Petroleum Institute has published API 579,
which is a comprehensive fitness-for-service guide that addresses various types of
flaws and damage, including cracks, general corrosion, local corrosion, pitting,
bulging, and weld misalignment.
The API 579 assessment of cracks implements the FAD method,
and is similar in many respects to CEGB R6, BS 7910, and SINTAP
Summary of International FAD-Based Procedures

145
API RP 579 uses engineering fracture mechanics to assess crack-like flaws.
A Failure Assessment Diagram (FAD) is applied to crack-like flaws, as illustrated below.
Linear elastic stress analysis is used to calculate the toughness ratio (Kr) and the
load ratio (Lr) for a component with a crack-like flaw.
Kr is the ratio of the linear elastic stress intensity factor (KI) to the material
fracture toughness (KMAT), while Lr is the ratio of the reference stress (σref) to
the material yield strength (σys).
For a given flaw and load, the value of Kr as a function of Lr is plotted on the FAD.
No failure is predicted for points below the failure assessment envelope,
whereas failure is likely to occur for points at or above the assessment envelope.
Calculations are repeated for other conditions to see where they fall with respect to
the envelope or to determine the critical conditions (a point on the failure envelope)
at which failure is predicted to be more likely to occur.
Failure Assessment Diagram (FAD) – API 579 Methodology

146
The API 579 procedure for evaluating cracks incorporates a failure assessment diagram (FAD)
methodology very similar to that in other documents, such as the British Energy R6 approach and the
BS 7910 method.
.
The basic assumption is that the flawed body could fail by one of two extreme failure modes - fracture
or plastic collapse (overload).
There are three different levels of FFS assessment:
- Level 1 FFS assessments (“Simplified assessment”) provide conservative screening criteria that
require the least amount of inspection and component information. Level 1 assessments usually do not
require extensive calculations. Either inspectors or plant engineers will conduct a Level 1 assessment.
- Level 2 FFS assessments (“Normal Assessment”) involve a more detailed evaluation of components
and usually require an accurate measurement of flaws or damage. Most Level 2 FFS assessments
require calculation of the required component thickness or of component stress. Either plant engineers
or engineering specialists will conduct level 2 assessments.
- Level 3 FFS assessments (“Ductile Tearing Instability”) require detailed evaluation of components.
Component flaws or damage must be accurately determined, and calculation methods often involve
numerical analysis such as the finite element method. Level 3 assessments often require the services of
engineering specialists experienced in advanced stress analysis, fracture mechanics, etc.
The assessment of the stability of the defect of this study is done through the use of the Failure
Assessment Diagram (FAD). On this diagram the assessment point is determined through the load
ratio and toughness ratio coordinates calculated according to the chosen level of assessment
The assessment of the stability of the defect of this study is done through the use of the Failure
Assessment Diagram (FAD). On this diagram the assessment point is determined through the load
ratio and toughness ratio coordinates calculated according to the chosen level of assessment.

147
Overview of an FFS Analysis for crack like flaws using the Failure Assessment
Diagram (FAD) – Level 2

148
API 579
• is supported by a number of organizations based in the USA where most experience resides.
• is designed at level 1 for use by plant inspectors and plant engineering personnel with the
minimum amount of information from inspection and about the component.
• covers a wide range of damage types typically found in refining and petrochemicals application,
and gives procedures for different types of metal loss, physical damage, low and high
temperatures, and crack like defects.
• is intended for equipment designed using the ASME code and materials and gives results
consistent with the original ASME design safety margins.
• may be used for equipment designed to other codes but users should be prepared to interpret the
procedures in an appropriate manner.
BS 7910
• was developed in the UK where TWI is the main source of expertise, training and software.
• is applicable to all metallic structures and materials and is written in a more generalized manner
without reference to a particular industry, design code or material thereby allowing users to decide
safety margins.
• requires some technical expertise in fracture mechanics and access to fracture parameter
solutions and toughness data at all levels.
• deals comprehensively with fatigue and fracture of defects in and around welded joints and gives
annexes covering advanced aspects such as mismatch, mixed mode loading, residual stress
effects and leak before break.
Synthesis - API 579 and BS 7910

149
Example - Defect Classified as a Longitudinal Crack
in a Pressurized C-Mn Steel Pipe
a = 2 mm
2c = 8 mm
2R = 1125 mm
t = 13 mm
Material: C-Mn Steel
The possible stresses are due to :
a) internal pressure
b) pipe self weight
c) point load (saddle)
d) overpressure due to water hammer
e) thermal expansion (negligible value)
Bending due to the pipe weight does not influence crack opening and for this reason
it will not be taken into account.
A stress of 129 MPa used in the calculation is determined by the sum of single stresses
taken at their highest admissible value.

150
Solution - API 579
Using a Level 2 assessment, proceed with the calculation in accordance with paragraph 9.4.3 of
Section 9 (assessment of crack-like flaws):
For the determination of the coordinates of the assessment point, it is necessary to calculate the Load
Ratio (Lr) and Toughness Ratio (Kr)
The standard requires characteristic input data, in order to assess the reference stress needed for the
calculation of (Lr).
Reference Stress:
where Pm and Pb are the Primary membrane stress and Primary bending stress, and Ms and g are
dimensionless coefficients affected by crack size.
Using the reference stress, it is possible to determine the dimensionless coefficient Lr , abscissa of the
FAD graph.
Lr is defined as:
For the determination of Kr ,two values of are calculated to take into account the possible instability
of the defect in the longitudinal and radial directions.

151
From which it is possible to determine Kr :
Taking into account the different values of

152
Now is it possible to assess a safety factor relative to the position of the assessment points:
Because the point at the intersection of the value of the abscissa and ordinate of the FAD graph
lies below the line of admissible defect, the crack in the pipe can be considered stable for an analysis
with invariant loads.

153
Solution – BS 7910
BS 7910 allows the application of a Level 2 assessment and assumes the same conditions of use as API 579
For the determination of the coordinates of the assessment point, it is necessary to calculate the Load
Ratio (Lr) and Toughness Ratio (Kr)
BS 7910 requires characteristic input data, in order to assess the reference stress needed for the calculation
of (Lr).
Reference Stress, σref :
Where Pm and Pb are the Primary membrane stress and Primary bending stress, Ms and α are dimensionless
coefficients influenced by the size of the crack.
The value of load ratio obtained from this assessment is:
.

154
Toughness Ratio (Kr) :
Where (Yσ) is calculated by means of factors presented in tables for different cases and sizes of the defect.
In this case it is necessary to evaluate the intensity factor for the deepest point (d) in the flaw and where the
flaw intersects the free surface(s).
Two Toughness Ratios result :
Section of the pipe with semi-elliptical
crack-like flaw and its parameters –
From Annex P BS 7910

155
Assessment points using BS 7910
The safety factor coefficients are then defined as follows:
Through an assessment of Level 2, the defect present in the pressurized pipe is safe under static loads.

156
The application of standards API 579 and BS 7910 has produced results in agreement that highlight in
particular the stability of the defect. Since the equation of the curve of the FAD graph is the same for
the two codes, it is possible to plot the results on the same graph:
Solution – Synthesis API 579 BS 7910
In blue are the values resulting from the application of API RP 579, in red those from BS 7910.
API 579 produces values slightly higher than those estimated by the standard BS 7910. This result is
influenced by the different methodology used to calculate the intensity factors.
Comparison between API RP 579
(blue cross) e BS 7910 (red circle)

157
There are no significant differences between the two standards for their application to this example,
however, BS 7910 has a wider range of applications than the one of API 579.
In particular, certain procedures developed in Europe (BS 7910, SINTAP, R6, ETM) are in many
aspects (mismatch, constraint, probabilistic aspects) even more advanced than the U.S. product.
However, the American standard API 579 is perhaps the code that better analyze the best
combination of damage mechanisms.
API 579 not only contains all the appropriate computational tools to support at the described
analysis in appendices, but also it is the only procedure which offers many reference curves for
the properties of materials used in the FFS analysis
Solution – Synthesis API 579 BS 7910 – Concluding Remarks

Environment assisted cracking (EAC) is
a process whereby cracking is induced
or accelerated by the presence of an
aggressive environment. It requires the
combination of a corrosive environment,
a susceptible material and a stress, and
includes what may be referred to as
stress corrosion cracking (SCC) and
corrosion fatigue.
SCC is defined as the cracking of metal
by the combined action of corrosion and
tensile stress (applied or residual), and
can cause a massive reduction in a
material’s strength with minimal material
loss.
SCC will not occur unless the applied K
is higher than a critical value, termed
KISCC. Above KISCC stress corrosion
cracks tend to grow at a constant rate
(assuming no changes to the local
stress-state or environmental conditions)
until they reach the material.’s fracture
toughness andfailure occurs. However,
the crack velocity is material-
environment specific.
159
Environment Assisted Cracking - SCC

Corrosion fatigue refers to damage or
failure of a material as a result of the
combined action of cyclic stresses
and a corrosive environment.
Depending on the particular material
environment combination and the
cyclic loading, this can result in higher
crack growth rates when compared to
standard mechanical fatigue in an
inert environment.
Because corrosion fatigue is
dependent on the specific material,
environment and load combination,
any quantitative modeling is
particularly complex, although various
approaches have been proposed.
160
Environment Assisted Cracking – Corrosion Fatigue
Types of corrosion fatigue crack growth

161
The underlying mechanism responsible for EAC will depend on the specific material environment system under
consideration. However, in broad terms, behavior can be attributable to either localized Anodic Dissolution or
hydrogen embrittlement (Cathodic Charging)
Many forms of SCC involve anodic dissolution as a key process, and this is usually localized along a susceptible
microstructural path such as grain boundaries. Anodic reaction processes are further categorized as follows:
• Slip-dissolution model.
• Anodic reaction induced cleavage (including film-induced cleavage and dissolution
enhanced plastic flow leading to cleavage).
• Surface mobility model.
Hydrogen embrittlement involves the absorption of hydrogen atoms onto the surface or into the bulk of a material
from the environment. There are many different mechanisms dependent on the material-environment combination
of interest. Hydrogen can be absorbed into all metals and the fact that it is a very small atom allows it to diffuse
more rapidly than larger atoms. Hydrogen tends to be attracted to regions of high triaxial tensile stress where the
metal structure is dilated (e.g. the regions ahead of crack tips or notches that are under stress) and
then assists fracture, possibly by making cleavage (brittle transgranular fracture) easier or possibly by assisting in
the development of intense local plastic deformation.
Work at TWI and elsewhere has shown that pipeline steels with much lower yield strengths can be susceptible to
some forms of embrittlement. Potential sources of hydrogen include:
• Welding;
• Process fluid/gases;
• Cathodic over-protection (in seawater);
• By-product of corrosion (CO2, H2S, crevice corrosion giving low pH).
Environment Assisted Cracking
Summary of SCC and Corrosion Fatigue

BS 7910 describes a flaw
acceptance criterion as follows:
where KI is the applied stress
intensity and fSCC is a factor of
safety, to be agreed between the
parties involved. This results in a
modified FAD, similar to that
illustrated, where for
KI<KISCC/fSCC the flaw is
tolerable, and no crack growth is
expected. However, when
KI>KISCC/fSCC the possibility
for stress corrosion crack growth
should be recognized.
162
FAD Approach to EAC - BS 7910
Failure assessment diagram incorporating cut-off for stress
corrosion cracking (after BS 7910, 2005 and FITNET, 2008, p9-11).

In such cases where crack
extension via SCC is not
tolerated, an alternative
methodology is to define the FAD
envelope in terms of .‘sour
service.’ properties, for example
using KISCC as the measure of
material fracture toughness.
This differs slightly from the
approach defined in documents
such as BS 7910 (2005), but is
believed to be conservative. This
approach is the subject of
ongoing research at TWI and
elsewhere.
163
FAD Approach to EAC - TWI
Failure assessment diagram envelope defined in terms of .‘sour service.’ properties
(e.g. using threshold stress intensity factor for stress corrosion cracking as the
measure of material fracture toughness).

165
Additional References
“NUMERICAL ANALYSIS OF CRACK PROPAGATION AND LIFETIME ESTIMATION”, Bo Ernst
Westergren Jensen, M.Sc. Thesis, Aalborg University, Esbjerg, 2015
“EFFECT OF WELDING RESIDUAL STRESS ON FRACTURE”, Xiaobo Ren, Doctorate Thesis,
Norwegian University of Science and Technology, 2010.
“STRUCTURAL INTEGRITY ASSESSMENT OF C-MN PIPELINE STEELS EXPOSED TO SOUR
ENVIRONMENTS”, Colum Holtam, Doctor of Engineering (EngD) Thesis, Loughborough
University, 2010.
“EFFECT OF WELDING RESIDUAL STRESS ON FRACTURE”, Xiaobo Ren, Doctorate Thesis,
Norwegian University of Science and Technology, 2010.
“FRACTURE MECHANICS Fundamentals and Applications”, T.L. Anderson, CRC Press
Taylor & Francis Group Pubs., 2005.
“RESIDUAL STRENGTH ANALYSIS OF LASER BEAM AND FRICTION STIR WELDED
ALUMINIUM PANELS FOR AEROSPACE APPLICATIONS”, Eduard Seib, Doctorate Thesis,
Technischen Universität Hamburg, 2005.

2-Fracture

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