FRACTURE & FATIGUE - Advanced Fracture Mechanics - O'Dowd (Notes)
107
Imperial College London Department of Mechanical Engineering ADVANCED FRACTURE MECHANICS Lectures on Fundamentals of Elastic, Elastic-Plastic and Creep Fracture 2002–2003 Course lecturer: Dr Noel O’Dowd
Transcript of FRACTURE & FATIGUE - Advanced Fracture Mechanics - O'Dowd (Notes)
Imperial College LondonDepartment of Mechanical Engineering
ADVANCED FRACTURE MECHANICSLectures on Fundamentals of Elastic,
Elastic-Plastic and Creep Fracture
2002–2003
Course lecturer: Dr Noel O’Dowd
CONTENTS
PageCourse Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii1. Linear Elastic Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Definition of energy release rate, G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Strain energy, energy release rate and compliance . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Stress analysis of cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Mixed mode fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.5 Concept of small scale yielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2. Non-linear Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.1 The J integral, (Rice, 1968) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2 Power law hardening materials—The HRR field . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.3 Crack tip opening displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.4 Relationship between J and G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.5 Evaluating J for test specimens and components . . . . . . . . . . . . . . . . . . . . . . . . . . 452.6 Application of non-linear fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.7 K dominance, J dominance and size requirements . . . . . . . . . . . . . . . . . . . . . . . . . 602.8 Standard test to determine JIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3. Micromechanisms for ductile and brittle fracture . . . . . . . . . . . . . . . . . . . . . . 663.1 Micromechanism of cleavage failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.2 Prediction of fracture toughness using the RKR model and the HRR field . . 673.3 Micromechanism of ductile failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.4 Prediction of fracture toughness using the MVC model and the HRR field . 693.5 Competition between brittle and ductile fracture . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4. Application of BS 7910 in failure assessments . . . . . . . . . . . . . . . . . . . . . . . . . 744.1 The failure assessment diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.2 Level 1 Failure Assessment Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.3 Level 2 Failure Assessment Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.4 Level 3 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5. Creep Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.1 Secondary creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.2 Estimation of C∗ in specimens and components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.3 Creep solutions for short times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.4 Characterisation of creep crack initiation and growth . . . . . . . . . . . . . . . . . . . . . . 895.5 Elastic-plastic creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.6 Micrographs of creep failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.1 Appendix A, Extracts from two key papers on non-linear fracture mechanics6.2 Appendix B, List of important equations for Advanced Fracture Mechanics6.3 Appendix C, Linear Elastic K field distributions
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September 2002
Imperial College LondonDepartment of Mechanical Engineering
4M/AME Advanced Fracture Mechanics
The course, which will be given by Dr O’Dowd, consists of approximately 22 lectures and9 tutorials. Examination will be by written paper at the end of the course and by problemsheets (5 in total) which will be distributed throughout the year and are worth 20% of thetotal course mark. 4M students must have taken the course 3M course, “Fundamentals ofFracture Mechanics”.
Aims
The principal aim of the course is to provide students with a comprehensive under-standing of the stress analysis and fracture mechanics concepts required for describingfailure in engineering components. In addition, the course will explain how to apply theseconcepts in a safety assessment analysis. The course deals with fracture under brittle, duc-tile and creep conditions. Lectures are presented on the underlying principles and exercisesprovided to give experience of solving practical problems.
Objectives
At the end of the course the student should be able to:1. Understand the mechanisms of fracture under brittle, ductile and creep conditions.2. Explain the relationship between linear elastic and non-linear fracture concepts and
the terms K, G, J and C∗.4. Establish the theoretical stress distributions ahead of a crack under brittle, ductile
and creep conditions.5. Appreciate how to make valid fracture toughness measurements on materials.6. Understand the theoretical basis behind fracture mechanics design codes and know
how to apply these codes to cracks in engineering components.
Relevant Textbooks
Most mechanics of materials textbooks provide an introduction to fracture mechanics,e.g. Mechanical Behaviour of Materials, by N.E. Dowling. The more advanced topics cov-ered in these lectures are dealt with by the following textbooks, which should be consideredbackground reading and are not required texts. The texts are listed in alphabetical order.
1. T.L. Anderson ‘Fracture mechanics: fundamentals and applications’, Edwards Arnold,London, 1991.
2. R.W. Hertzberg, ‘Deformation and fracture mechanics of engineering materials’, Wileyand Sons, New York, 1989.
3. M.F. Kanninen and C.H. Popelar, ‘Advanced fracture mechanics’, Oxford UniversityPress, 1985.
4. G.A. Webster and R.A. Ainsworth, ‘High temperature component life assessment’,Chapman and Hall, London, 1994.
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September 2002
Imperial College of Science Technology and Medicine,
Department of Mechanical Engineering
4M/AME Advanced Fracture Mechanics
Introduction
Fracture mechanics concerns the design and analysis of structures which contain cracks
or flaws. On some size-scale all materials contain flaws either microscopic, due to cracked
inclusions, debonded fibres etc., or macroscopic, due to corrosion, fatigue, welding flaws
etc. Thus fracture mechanics is involved in any detailed design or safety assessment of
a structure. As cracks can grow during service due to e.g. fatigue, fracture mechanics
assessments are required throughout the life of a structure or component, not just at start
of life. Fracture mechanics answers the questions: What is the largest sized crack that a
structure can contain or the largest load the structure can bear for failure to be avoided?
How long before a crack which was safe becomes unsafe? What material should be used
in a certain application to ensure safety?
Studies in the US in the 1970s by the US National Bureau of Standards estimated
that “cost of fracture” due to accidents, overdesign of structures, inspection costs, repair
and replacement was on the order of 120 billion dollars a year. While fracture cannot
of course be avoided, they estimated that, if best fracture control technology at the time
was applied, 35 billion dollars could be saved annually. This indicates the importance of
fracture mechanics to modern industrialised society.
The topics of linear elastic fracture mechanics, elastic-plastic fracture mechanics and
high temperature fracture mechanics (creep crack growth) are dealt with in this course.
The energy release rate method of characterising fracture is introduced and the K and
HRR fields which characterise the crack tip fields under elastic and plastic/creep fracture
respectively are derived. The principal mechanisms of fracture which control failure in the
different regimes are also discussed. In the later part of the course, the application of these
fracture mechanics principles in the assessment of the safety of components or structures
with flaws through the use of standardised procedures is discussed.
The approach taken in this course is somewhat different from that in Fundamentals
of Fracture Mechanics (FFM) as here more emphasis is put on the mechanics involved and
outlines of mathematical proofs of some of the fundamental fracture mechanics relation-
ships are provided. There is some revision of the topics covered in FFM, particularly in
the area of linear elastic fracture mechanics though the approach is a little different.
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1. Linear elastic fracture mechanics
1.1 Definition of energy release rate, GGriffith (1924) derived a criterion for crack growth using an energy approach. It is based
on the concept that energy must be conserved in all processes. He proposed that when
a crack grows the change (decrease) in the potential energy stored in the system, U , is
balanced by the change (increase) in surface energy, S, due to the creation of new crack
faces.
( )∆S As= γ δ
∆ a
B
Figure 1.1, Schematic of a crack growing by an amount ∆a.
Consider a through-thickness crack in a body of thickness B (see Fig. 1.1). For
fracture to occur energy must be conserved so,
∆U + ∆S = 0.
The change in surface energy, ∆S = δAγs where δA is the new surface area created
and γs is the surface energy per unit area, as illustrated above. The change in area
δA = 2B∆a, (the factor of 2 arises because there are two crack faces).
Inserting these values and dividing across by B∆a we get
− 1B
∆U
∆a= 2γs.
Rewriting as a partial derivitive we get Griffith’s relation,
− 1B
∂U
∂a= 2γs
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If this equation is satisfied then crack growth will occur. The energy release rate, G is
defined as
G = − 1B
∂U
∂a.
In almost all situations ∂U/∂a is negative, i.e. when the crack grows the potential
energy decreases, so G is positive. Note that the 1/B term is often left out and U is
then the potential energy per unit thickness. G has units J/m2, N/m or MPa·m and is
the amount of energy released per unit crack growth per unit thickness. It is a measure
of the energy provided by the system to grow the crack and depends on the material,
the geometry and the loading of the system. The surface energy, γs, depends only on
the material and environment, e.g. temperature, pressure etc., and not on loading or
crack geometry.
From the above, a crack will extend when
G︸︷︷︸crack driving force
≥ 2γs = Gc︸︷︷︸material toughness
.
It was found that while Griffith’s theory worked well for very brittle materials such as
glass it could not be used for more ductile materials such as metals or polymers. The
amount of energy required for crack was found to be much greater than 2γs for most
engineering materials. The result was therefore only of academic interest and not much
attention was paid to this work outside the academic community.
In 1948 Irwin and Orowan independently proposed an extension to the Griffith
theory, whereby the total energy required for crack growth is made up of surface energy
and irreversible plastic work close to the crack tip:
γ = γs + γp,
where γp is the plastic work dissipated in the material per unit crack surface area created
(in general γp >> γp). Then the criterion for fracture becomes
− 1B
∂U
∂a≥ 2(γs + γp)
or
G = − 1B
∂U
∂a≥ 2(γs + γp) = Gc.
The Griffith and Irwin/Orowan approaches are mathematically equivalent, the only
difference is in the interpretation of the material toughness Gc. In general Gc is obtained
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directly from fracture tests which will be discussed later and not from values of γs and
γp. The critical energy release rate, Gc, can be considered to be a material property
like Youngs Modulus or yield stress. It does not depend on the nature of loading of the
crack or the crack shape, but will depend on things like temperature, environment etc.
We next examine how to determine the potential energy and the energy release rate for
a linear elastic material.
1.2 Strain energy, energy release rate and compliance
The energy release rate G can be written in terms of the elastic (or elastic-plastic)
compliance of a body. Before showing this, some general definitions will be given.
1.2.1 Strain energy density
Strain energy density, W , is given by,
W =∫ ε
0
σdε,
where σ is the stress tensor (matrix), ε is the strain tensor (matrix). Under uniax-
ial loading W is simply the area under the stress-strain curve (note: not the load-
displacement curve)as illustrated in Fig. 1.2. In general, the strain energy will not be
constant throughout the body but will depend on position.
ε�
σ�
W
σ, ε����
Figure 1.2, Definition of strain energy density W under uniaxial loading.
1.2.2 Elastic and plastic materials
For an elastic material all energy is recovered on unloading. For a plastic material,
energy is dissipated.
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The strain energy density of an elastic material depends only on the current strain,
while for a plastic material W depends on loading/unloading history. If the material is
under continuous loading, W for an elastic and plastic material are the same. However,
if there is total or partial unloading there is a difference. The response of an elastic
and an elastic-plastic material such as steel is shown in Fig. 1.3. The elastic material
unloads back along the loading path, i.e. no work is done in a cycle which in fact is
the definition of an elastic material. For an elastic-plastic material there is generally an
initial elastic regime where the stress-strain curve is linear (stress directly proportional
to strain) and energy is recoverable and a nonlinear plastic regime where energy is
dissipated (unrecoverable). Unloading is usually taken to follow the slope of the initial
elastic region. There is then a permanent plastic deformation and the work done in a
cycle is given by the area under the curve.
unloading
loading
unloading
σ�
Work done = 0Elastic Material
ε�
loadingσ�
ε�
Elastic-plasticmaterial
Work done
Figure 1.3, Comparison between behaviour of an elastic material (left) and an elastic-
plastic material (right).
An elastic material need not be a linear elastic material—there are elastic materials, e.g.
rubbers, which are non-linear. However, the term elastic is often used as a shorthand
for linear elastic. For a linear elastic material,
σ = Dε,
where D is the elasticity matrix and σ and ε are the stress and strain matrices.
W =12σε.
In uniaxial loading
W =σ2
2E.
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Power law hardening, a non-linear stress-strain law where strain is proportional to stress
raised to a power, is often used to represent the plastic behaviour of materials
σ = Dε1/n,
where again D is a matrix of material constants and n is the strain hardening exponent,
1 ≤ n ≤ ∞. In this case, it can be shown that,
W =n
n + 1σε.
1.2.3 Definition of Strain energy, Ue
The strain energy of the body is a measure of how much strain energy is stored in the
body, depends on the material and loading and is given by,
Ue =∫
V
WdV ,
where V is the volume of the body. The strain energy, Ue, is the strain energy density
integrated over the whole body and it can be shown that it is equal to the area under
the load-displacement curve. For a linear elastic material the strain energy is simply,
Ue =P∆2
,
where P and ∆ are the applied load and conjugate displacement. For a power law
hardening material it can be shown that,
Ue =n
n + 1P∆.
1.2.4 Definition of Potential energy, U :
The potential energy is made up of the internal strain energy and the external work
done on the body and depends on the way the body is loaded.
∆
Figure 1.4, Schematic of a loaded cracked body.
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For the body shown in Fig. 1.4, the potential energy will have a different definition
depending on whether it is loaded by a prescribed load or a prescribed displacement:
For prescribed displacement, ∆: U = Ue
For prescribed load, P : U = Ue − P∆
Prescribed load means that the load will be fixed (constant) during an increment of
crack growth, while prescribed displacement means that displacement is fixed during
crack growth. For prescribed displacement, no work can be done by the external loading
during crack growth because the displacement remains fixed (work = force × displace-
ment) so the change in potential energy is only due to the change in strain energy.
1.2.5 Definition of Compliance, C:
The compliance of a body is the inverse of the stiffness. It is not a material property,
but depends on the loading and geometry. For a linear elastic body with a crack of
length a we can write
∆ = C(a)P,
where C(a) is the compliance and depends on geometry (including crack length, a),
Youngs modulus, E and ν. For a power law hardening material we can write
∆ = C(a, n)Pn,
where the compliance C(a, n) also depends on the hardening exponent.
1.2.6 Derivation of G from compliance for linear elastic material:
For fixed load:
U =P∆2− P∆ = −P∆
2
G = − 1B
(∂U
∂a
)
P
=P
2B
(∂∆∂a
)
P
where the notation(
∂
∂a
)
P
emphasises that load P is held constant.
∆ = C(a)P
⇒(
∂∆∂a
)
P
= PdC
da
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Therefore
G =1
2BP 2 dC
da.
If displacement ∆ is held fixed:
U = Ue =P∆2
and it can be shown (this is left as an exercise) that again
G =1
2BP 2 dC
da.
In other words although the potential energy depends on mode of loading, the
energy release rate, G, does not and is independent of the nature of loading whether
by imposed displacement or imposed load. (This is true in general not only for linear
elastic materials.)
1.2.7 Stability of Crack Growth
If we examine the above equation for G and differentiate again with respect to a, we get
for fixed load: (∂G∂a
)
P
=1
2BP 2 d2C
da2
For fixed displacement we get
(∂G∂a
)
∆
=1
2BP 2
(d2C
da2− 2
C
(dC
da
)2)
.
For most geometries, d2C/da2 is positive, so for prescribed load G increases with
crack growth. Therefore crack growth is unstable—an increase in a leads to an increase
in G. For prescribed displacement, if 2/C(dC/da)2 > d2C/da2 then
∂G∂a
< 0
so G decreases with crack growth and crack growth is stable, i.e. the applied displace-
ment must be increased to maintain crack growth.
Criteria for unstable crack growth:
G = Gc and∂G∂a
> 0.
Stability arguments are important because while an amount of stable crack growth may
be acceptable, unstable fracture must always be avoided.
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Most materials are ductile and even under nominally linear elastic conditions, frac-
ture toughness increases with crack growth as shown. This is what is known as a
resistance curve or sometimes an R curve (see Fig. 1.5).
Gr
∆a
Gc
R curve
No R curve behaviour
Figure 1.5, Schematic of material resistance curve behaviour
In these circumstances, satisfying the above criteria will not necessarily lead to unstable
crack growth and the criteria for unstable crack growth are
G = Gr and∂G∂a
>dGr
da.
1.3 Stress analysis of cracks
1.3.1 The K-field for linear elastic materials, Irwin (1958)
An alternative approach to the energy approach in the analysis of cracks is the
‘stress intensity’ approach where the stress and strain field at the crack tip are examined.
In many situations the energy and stress intensity approaches are equivalent and give
the same predictions. However, it is important to be familiar with both approaches.The
energy approach is appropriate mainly for elastic (linear or non-linear elastic) materials.
The stress intensity approach is perhaps more flexible and can be applied to a wider
range of materials.
We wish to determine the stress state at the crack tip. The most general 3-D stress
state is,
σ(x, y, z) =
σxx σxy σxz
σyx σyy σyz
σzx σzy σzz
,
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i.e. six independent stress components and all components depend on x, y and z. To
simplify we first assume an infinitely sharp, straight crack front and align the axes along
the crack front (see Fig. 1.6).
z
x
y
Figure 1.6 Schematic of a three dimensional crack
Assume that as the crack tip is approached the stress variation along the crack front
(in the z direction) is negligible compared to the variation in the x and y direction,
∂σxx
∂z=
∂σyy
∂z=
∂σzz
∂z. . . = 0.
Therefore close to the crack tip the stresses and strains do not depend on z—the
stresses behave as if responding to a 2-D deformation field. The stress state at a crack
tip is therefore a combination of plane stress/strain, u(x, y), v(x, y), and an anti-plane
strain problem, w(x, y), where u and v are the x and y (in plane) displacements, and
w is the out of plane (z) displacement. For a linear elastic problem, we may examine
these modes separately and determine the total stress by (linear) superposition.
1.3.2 The plane stress/strain crack tip fields
For simplicity we use polar coordinates as shown in Fig. 1.7.
θθ
θ
σσ
Figure 1.7 Polar coordinates centered at a sharp crack tip
We focus attention on the crack tip so we do not consider the remote boundary condi-
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tions. The only relevant boundary conditions are the stress free crack faces:σθθ(r,±π) = 0
τrθ(r,±π) = 0.
For a linear elastic material, this problem may be solved by using the Airy stress function
method (covered in the Advanced Stress Analysis course) which gives an exact solution
to linear elastic stress problems. The details of the proof are beyond the scope of the
course, but an outline of the steps in the proof is given below.
The stress field is represented as an infinite series in r, i.e.
σ(r, θ) =∞∑
i=1
Airλifi(θ)
where Ai and λi are unknown real constants and fi are unknown functions of θ. If suffi-
cient terms are taken the exact solution to any problem can be obtained. However, for a
fracture mechanics analysis, we focus on the terms which make the largest contribution
to the stress in the vicinity of the crack tip, r → 0.
Values of λ ≥ 0 can be ignored, as the stress term corresponding to this value of λ
tends to zero as we approach the crack tip (i.e. rλ → 0 when r → 0 for λ > 0; rλ = 1
when λ = 0). It can also be shown that in order to have bounded strain energy at the
crack tip terms of order lower than −1 must also be excluded. (Physical requirements
rule out infinite strain energy, though infinite stress is allowed).
It can then be shown that the only value of λ, between −1 and 0, which satisfies
equilibrium and the above boundary conditions is λ = −1/2, i.e.
σ ∼ A√rf(θ) + . . .
The dots indicate that this is the first (and most important) term in a series for the
crack tip stress field.
Conventionally, the arbitrary constant A is replaced by the ‘stress intensity factor’
K and the solution is divided into Mode I (tension) and Mode II (shear) solutions. The
Mode I stresses at the crack tip can be derived (the proof is relatively straightforward)
and are as follows:
σθθ =KI√2πr
cos3θ
2+ . . .
σrr =KI√2πr
(cos
θ
2+ cos
θ
2sin2 θ
2
)+ . . .
τrθ =KI√2πr
sinθ
2cos2
θ
2+ . . .
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Note that at θ = 0,
σθθ = σrr =KI√2πr
; τrθ = 0;
and at θ = ±π (the crack faces),
σθθ(r,±π) = 0
τrθ(r,±π) = 0.
For Mode II the result is:
σθθ =KII√2πr
(−3 sin
θ
2cos2
θ
2
)+ . . .
σrr =KII√2πr
(sin
θ
2− 3 sin3 θ
2
)+ . . .
τrθ =KII√2πr
(cos
θ
2− 3 cos
θ
2sin2 θ
2
)+ . . .
and at θ = 0
σθθ = σrr = 0; τrθ =KII√2πr
;
and again at θ = ±π,σθθ(r,±π) = 0
τrθ(r,±π) = 0.
The value of the out of plane stress, σzz, will depend on whether we assume plane stress
or plane strain,
σzz ={
ν(σrr + σθθ) plane strain0 plane stress
In a real 3-D geometry, σzz will vary from plane strain at the centre to plane stress at
the surface. However, the other stress components remains the same.
For Mode III antiplane shear mode we get
τzθ =KIII√
2πrcos
θ
2+ . . .
τzr =KIII√
2πrsin
θ
2+ . . .
and all other stress components are zero.
These stress fields fully describe the stress fields for a sharp crack in a linear elastic
material. The values of KI , KII and KIII are undetermined by the above analysis
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and will be found by considering the remote boundary conditions. The term singular
(where singular means infinite) fields is often used to describe these fields as they predict
infinite stress stress at r = 0. The singularity is referred to as a square root singularity,
because, σ ∼ 1/√
r.
The three modes are shown in Fig. 1.8 and the full mathematical description of the
three modes is provided in Appendix C.
Y
X
Z
Y
X
Z
Y
X Z
Mode I Mode II Mode III
Mode I: Opening or Tensile Mode
Mode II: Sliding or In-Plane Shear Mode
Mode III: Tearing or Anti-Plane Shear Mode
Figure 1.8 The three modes of crack tip deformation
Note: The K field is not the exact solution to the stress field in a cracked body. It is the
solution to the stress field as we approach the crack tip, where the approximations used
in the derivation of the K field apply. Because this term in the stress field is so much
larger than the other terms, we are justified in neglecting them and a single parameter
K then defines the stress at the crack tip (more on this point later.)
1.3.4 Crack tip opening displacement
The crack tip opening displacement, ∆u in Fig. 1.9, can be obtained directly from the
K field.
12
∆θ
Figure 1.9 Definition of crack tip opening displacement, ∆u
Mathematically, the crack tip opening displacement is defined as,
∆u(r) = uy(r, θ = π)− uy(r, θ = −π).
In the above equation uy is the y displacement and the equation for this is given in
Appendix C. Inserting the equation for uy from Appendix C gives for Mode I,
∆u(r) = 2KI
E
√r
2π(1 + ν)(κ + 1),
where
κ =
3− 4ν plane strain
3− ν
1 + νplane stress.
Substituting for κ the above can be rewritten as
∆u(r) = 8KI
E′
√r
2π
where
E′ =
E
1− ν2plane strain
E plane stress.
The crack tip opening displacement, as defined here, is not a single number but gives
the relative displacement of the crack faces at a distance r from the crack tip, as defined
by the K field. Note that
σ, ε ∝ 1√r; u, CTOD ∝ √
r,
where r measures distance from the crack tip.
The use of the CTOD (crack tip opening displacement) as a fracture parameter is
based on non-linear elastic fracture mechanics and will be discussed later.
13
1.3.5 K in infinite and finite bodies
For some idealised geometries an exact value for KI , KII , KIII can be determined.
Consider an infinite plate, with a straight through thickness crack of length 2a, loaded
at infinity by tension, σ∞yy, and shear, τ∞xy and τ∞yz as shown in Fig. 1.10.
τ
τσ
xy
yz
yy
Figure 1.10 An infinite plate with a centre crack of length 2a.
It can be shown that,KI =
√πσ∞yy
√a
KII =√
πτ∞xy
√a
KIII =√
πτ∞yz
√a
This is an exact solution for K and can be determined analytically. Note that a
σxx stress applied at infinity has no effect on K since it is parallel to the crack. Note
also that the modes are uncoupled, i.e. τ∞xy does not contribute to KI or σ∞yy to KII
etc. For a homogenous material, the crack modes are always uncoupled—it is somewhat
more complicated for an interface crack (a crack lying at the interface of two different
materials).
For an edge crack of length a in an infinite plate we get
KI = 1.12√
πσ∞yy
√a
For an infinite plate, (a << W ) with an edge crack under pure moment M (Fig. 1.11)
we get,
W
a
Figure 1.11 Infinite plate with an edge crack under bending
14
KI = 1.12√
πσ∞b√
a,
where σ∞b is the bending stress,
σ∞b =My
I= M
W
212
W 3b=
6M
W 2b,
with W the width of the plate and b the thickness.
For a finite sized plate of width 2W with a center crack under tension and shear,
KI = YI(a/W )σ∞y√
a
KII = YII(a/W )τ∞xy
√a
KIII = YIII(a/W )τ∞yz
√a
where YI , YII , YIII are the shape factors (and clearly as a/W → 0, Y → √π).
The values of YI , YII , YIII must generally be obtained numerically, e.g. from a
finite element solution and have been tabulated in handbooks for many geometries, e.g.
Sih, G. C. Handbook of Stress-Intensity factors, 1974.
1.3.5 Comparison of K field with full stress field
As pointed out in Section 1.3.1 the K field is not the full solution to the stress field in
a cracked body. It is the solution to the stress field as we approach the crack tip, where
the assumptions used in the derivation of the K field apply. Consider the infinite plate,
with a crack of length 2a (Fig. 1.12). The exact 2-D solution to the stress field can be
obtained using stress functions.
2a
Figure 1.12 Infinite plate with a centre crack under remote tension
15
With only tension applied remotely, the σyy stress ahead of the crack along the center-
line, y = 0, is given by
σyy =σ∞yy |x|√x2 − a2
for |x− a| > 0, (∗)
where x measures distance from the centre of the plate. Replacing x by r where r
measures distance from the crack tip, i.e.
r = x− a ⇒ x = r + a,
gives
σyy =σ∞yy |r + a|√(r + a)2 − a2
=σ∞yy |r + a|√
r2 + 2ar + a2 − a2=
σ∞yy |r + a|√r2 + 2ar
.
Focusing on the crack tip, by letting r → 0; then r + a → a and r2 + 2ar → 2ar and we
get
σyy ≈σ∞yy a√
2ar=
σ∞yy
√a√
2r.
If we compare this result with the K field for a Mode I crack derived earlier,
σyy =KI√2πr
.
Equating the two expressions for σyy we get, KI =√
πσ∞yy
√a the KI value for this
geometry.
The comparison between the ‘exact’ stress field from equation * above and the K
field is shown on a log-log plot in Fig. 1.13(a). It is seen that for r < 0.1a the K field
and the ‘exact’ stress distribution are in excellent agreement.
For most other geometries the stress fields must be calculated numerically. The
stress field obtained for a center cracked panel with a/W = 0.5 loaded under tension,
calculated from an FE analysis, compared with the K field for this geometry is also
shown in Fig. 1.13(b) (the Y value for the geometry is 1.15√
π.)
If the K field agrees with the ‘actual’ stress field over a reasonable distance the
specimen is said to be K dominant. Most standard test specimens are K dominant.
However, the commonly used DCB (double cantilever beam) has a small zone of K
dominance and results from this specimen are generally interpreted using the energy
release rate rather than the stress intensity factor. Plasticity can further affect the zone
of K dominance and we will have more on that later.
16
Figure 1.13 Comparison between exact solution and K field, (a) infinite plate with a
centre crack, (b) centre cracked plate with a/W = 0.5.
Note that the analysis outlined in this section is one way of obtaining the value of K.
If the full stress field for a geometry is known, the value of K may be obtained via
KI = limr→0
√2πr [σyy(θ = 0)]
and similarily
KII = limr→0
√2πr [σxy(θ = 0)]
This is a rather cumbersome way of determining K and there are many other simpler
numerical methods which can be used. The simplest method is probably the use of the
J integral which can be converted to K as we will see later.
1.3.5 Connection between G and K
Consider the strain energy released when crack grows an amount ∆a, i.e. go from state
A to state B in Fig. 1.14. The stress normal to the crack, σyy, relaxes from σyy(x) to
zero over ∆a while displacement from increases from 0 to ∆u.
17
Y
X
∆u
∆a
σyy
X�
∆a
Energyreleased
u ∆u
State A
State B σyy
Figure 1.14 Process of crack growth from State A to State B
(Alternatively, one could consider the work required (i.e. energy absorbed) to close the
crack an amount ∆a—the result is the same.)
Energy released (per unit thickness) =∫ ∆a
0
12σyy(x)∆u(x)dx
where ∆u(x) is the separation of the crack faces, the crack opening displacement.
For Mode I:
σyy(x) =K√2πx
(dropping the subscript ‘I’ for convenience) As shown in Fig. 1.14, the crack opening
displacement is measured from the new position of the crack tip, i.e. with axis X ′,
∆u(x′) =8K(a + ∆a)
E′
√−x′
2π,
(−x′, because in the equation for crack opening displacement, the distance to the crack
tip, r, is always positive.)
Since x′ = x−∆a we can write,
∆u(x) =8K(a + ∆a)
E′
√∆a− x
2π,
18
and substituting in the above equation we get
Energy released =2K(a) K(a + ∆a)
πE′
∫ ∆a
0
√∆a− x
xdx.
Now by definition, energy released (per unit thickness)= G∆a, therefore,
G∆a =2K(a)K(a + ∆a)
πE′
∫ ∆a
0
√∆a− x
xdx.
Evaluating the integral, ∫ ∆a
0
√∆a− x
xdx =
π
2∆a,
and the equation becomes,
G∆a =K(a)K(a + ∆a)
E′ ∆a.
Dividing by ∆a and letting ∆a → 0 so K(a + ∆a) → K(a) = K we get
G =K2
E′ ,
where (as before)
E′ =
E
1− ν2plane strain
E plane stress.
So there is an one-to-one relationship between G and K for a linear elastic material.
Therefore for a linear elastic material, the energy approach using G and the stress
intensity approach using K are equivalent. When G reaches its critical value Gc then K
must reach its critical value Kc.
The above analysis is for Mode I loading, K = KI . The more general relationship
for KI , KII , KIII which can be proven by an identical approach is,
G =K2
I + K2II
E′ +K2
III
2G
where G is the shear modulus.
1.3.6 Fracture toughness, KIC
Near the crack tip, r → 0, the stress and strain fields are well described by K. If
two geometries have the same value of K then they have the same associated stress
(and strain) field. This is known as the concept of ‘similitude’—K contains all the
19
information about loading and crack geometry. Thus, if we have a Mode I test specimen
which fractures at a load with stress intensity KI we designate this value to be KIC . If
an engineering structure with a crack is subjected to a loading which leads to a stress
intensity factor KI ≥ KIC then fracture will occur in the structure.
1.3.7 Stress based criterion for fracture
The stress at the crack tip is infinite regardless of the magnitude of K. Therefore to
formulate a stress based fracture criterion it is necessary to incorporate a distance. The
fracture criterion is then phrased in terms of the attainment of a critical stress at a
critical distance. A typical distance used is on the order of a grain size, say 10 µm. For
example if σc = 500MPa and rc = 10µm, then under Mode I loading
σyy =KI√2πr
.
At failure
σyy = σc| at r = rc=
KIC√2πrc
⇒ KIC = σc
√2πrc = 4MPa · √m
The use of a critical stress at a critical distance is often called the RKR model, after
Ritchie, Knott and Rice who first proposed it. (See also the extract from the paper by
Ritchie and Thompson, Appendix A.)
Brittle, stress induced failure is known as cleavage. An alternative failure mode is
ductile tearing which is associated with large plastic crack tip strains. Note that it is not
necessary to know the mechanism of fracture in order to predict whether fracture will
occur. For the engineer, the principal motivation behind fracture mechanics is to develop
(reasonably) simple methods to predict fracture not to analyse micromechanisms of
failure. Mechanisms of fracture are discussed in more detail later in the course.
1.4 Mixed mode fracture mechanics
1.4.1 Mode I and Mode II testing
The majority of fracture testing is carried out under Mode I conditions as this is gener-
ally the critical mode for failure. However, increasingly mixed mode fracture toughness
tests (combination of Mode I and Mode II) are being carried out to cover the full range
of a materials response to mechanical load.
20
At crack tip: bending moment only, no shear=> pure Mode I, M = 0
A B
B A
SFD
BMD
At crack tip: shear force only, no bending=> pure Mode II, M = 1
SFD
BMD
Typical test specimens for testing in Mode I and Mode II are shown in Fig. 1.15 and
1.16 respectively. By varying the position of the specimen relative to the applied load,
the asymmetric bend specimen in Fig. 1.16 can also be used for mixed mode testing.
1.4.2 Definition of Mode-Mixity
For a 2-D crack under mixed mode (combination of tension and shear) loading the stress
ahead of the crack is given by
σyy =KI√2πr
; τxy =KII√2πr
.
Mode mixity is a measure of the ratio of Mode I to Mode II. It is usually expressed as
21
an angle φ (the phase angle) where
φ = tan−1 KII
KI
φ = 0 ⇒ Mode I φ = π/2 ⇒ Mode II (since tan(π/2)= ∞)
or sometimes as M where
M =2π
φ
M = 0 ⇒ Mode I M = 1;⇒ Mode II
1.4.3 Crack path under mixed mode loading
It has been observed that a crack subjected to mixed mode loading (Figure 1.17) will
often not follow a straight path but will branch out of the plane. This has been observed
for ductile metals and ceramics.
θ
xy
σyy
σ
Figure 1.17, Shear and normal stresses for a crack tip under mixed mode loading
1.4.4 Criteria for crack kinking
A number of criteria which demonstrate good agreement with experimental observations
have been suggested. These give very similar predictions but are subtly different in the
way the problem is formulated mathematically.
1. Crack branches in direction of maximum hoop (σθθ) stress (stress based criterion)
2. Crack branches in direction of maximum energy release rate. (Griffith Criterion)
3. Crack branches in direction of local zero Mode II, in direction such that kII = 0
1.4.5 Maximum hoop stress criterion
The first criterion is the simplest to employ. The hoop stress field is given by (see
Appendix C)
σθ =KI√2πr
cos3 (θ/2) +KII√2πr
[−3 sin (θ/2) cos2 (θ/2)].
22
Note that the hoop stress is normal to any branched crack.
θ
σθ
Figure 1.18 Hoop stress, σθθ at an angle θ
For the maximum hoop stress criterion the branching angle, θ, is then given by the
angle which satisfies∂σθ
∂θ= 0
i.e.
KIdfθθ
1 (θ)dθ
+ KIIdfθθ
II (θ)dθ
= 0
where fθθ1 (θ) = cos3 (θ/2) and fθθ
II (θ) = −3 sin (θ/2) cos2 (θ/2). Therefore, given KI
and KII we can solve for θ. The solution for θ proved in Section 1.4.8 is given by:
tan (θ/2) =14
KI
KII−
√(KI
KII
)2
+ 8
for KII 6= 0.
Note that since we assume that the critical distance is the same for all angles, a distance
rc does not enter into the solution for the branching angle.
1.4.6 Griffith criterion and kII = 0 criterion
The other two criteria are more difficult to solve analytically. The approach taken
is to consider the actual branch and calculate the new value of KI and KII (designated
kI and kII) for the branched crack (see Fig. 1.19).
θ
k kI II,K KI II,
Figure 1.19 Local stress intensity factors, kI and kII ahead of a branching crack.
It can be shown that the local kI and kII are given by:
kI(θ) = a(θ)KI + b(θ)KII
23
kII(θ) = c(θ)KI + d(θ)KII
where, as indicated, a, b, c and d depend on the angle θ. Analytical solutions have not
been obtained for these values so numerical solutions have been used. The two criteria
for choosing the crack path, are then either based on the path which makes kII(θ) = 0
or the path which maximises G(θ). For the second case,
G(θ) =kI(θ)2 + kII(θ)2
E′
Problem is to maximize G(θ) which is done numerically. (Note that the usual Gonly gives energy for crack growth in the plane of the crack, i.e. for θ = 0. Under mixed
mode conditions it may be energetically more favourable for the crack to grow at some
other angle.)
A comparison between the three theories is shown in Fig. 1.20. Note that for phase
angle M = 1 (pure Mode II), θ ≈ 70◦ for maximum hoop stress theory and θ ≈ 80◦
for kII = 0 theory. Also included is some experimental data for alumina (a brittle
ceramic). The maximum hoop stress theory seems to give the best agreement with the
experimental data though there is significant scatter in the data.
Figure 1.20, Branching angle, θ, based on three criteria—comparison with experimental
data (from Suresh et al., J. American Ceramics Soc. 1990).
1.4.7 Dependence of fracture toughness on mode mixity
The previous section discussed how to determine the angle for crack growth under mixed
mode loading. More importantly we can also determine the dependence of the fracture
24
toughness on mode mixity. As stated earlier, the material parameters, GC or KIC apply
to Mode I only, so it is of interest to examine mixed mode conditions.
Of course, one can do this experimentally, simply by testing a specimen under different
mode mixities, using for example the symmetric and asymmetric bend specimens, in
Figs. 1.15 and 1.16, and determine the values of KI and KII at fracture. The advantage
of theoretical models is that they provide insight into the fracture process and assist in
extrapolating from one situation to another without the need for additional testing.
Consider again the hoop stress criterion. Assume that the branching angle has
been determined by the earlier analysis and is designated θ, then,
σθ(θ) =1√2πr
(KIfI(θ) + KIIfII(θ)
)
Fracture occurs when the hoop stress reaches a critical value, σc, at a critical distance
rc so we get1√2πrc
(KIfI(θ) + KIIfII(θ)
)= σc
KIC is the fracture toughness when KII = 0, θ = 0, fI(θ) = 1, therefore
√2πrc σc = KIC .
Substituting, we get,KI
KICfI(θ) +
KII
KICfII(θ) = 1
This is a universal curve with only one material parameter, KIC , which can be obtained
from a single experiment. This is analagous to the concept of a yield surface in plasticity
where only a single yield stress needs to be determined, rather than a set of values for
every load condition. Figure 1.21 shows the fracture toughness curve data for alumina
along with the experimental data. As always with ceramics there is a lot of scatter in
the data but the trend is quite well captured by the maximum hoop stress theory, or a
maximum energy theory (Griffith theory).
Of course, the validity of the expression for fracture toughness depends on whether
the assumptions of the models are correct—while either the Griffith or maximum hoop
stress theory works well in this case, neither may hold for another material.
25
Figure 1.21, Mixed mode fracture toughness locus for alumina, (from Suresh et al., J.
American Ceramics Soc. 1990)
26
1.4.8 Derivation of branching angle based on critical hoop stress
σθ =1√2πr
(KIfI(θ) + KIIfII(θ))
For a given KI and KII we seek to find the angle θ at which the hoop stress, σθ is a
maximum. The r dependence is the same at all angles and is given by the square root
singularity but the amplitude depends on the angle, θ.
fI(θ) = cos3θ
2; fII(θ) = −3 sin
θ
2cos2
θ
2
Criterion is :∂σθθ
∂θ= 0 ⇒ KIf
′I(θ) + KIIf
′II(θ) = 0
where ′ denotes differentiation. Denote the solution to this equation to be θ. i.e.
KI
(−3
2cos2
θ
2sin
θ
2
)+ KII
(3 sin
θ
2
[cos
θ
2sin
θ
2
]− 3
2cos3
θ
2
)= 0
⇒ 32KI cos2
θ
2sin
θ
2= 3KII cos
θ
2
[sin2 θ
2− 1
2cos2
θ
2
]
Now
sin2 θ
2− 1
2cos2
θ
2=
14(1− 3 cos θ)
and above becomes
KI cosθ
2sin
θ
2=
12KII(1− 3 cos θ)
and since
sin 2α = 2 sin α cosα
we rewrite this as
KI sin θ = KII(1− 3 cos θ)
Thus branching angle θ satisfies
sin θ
1− 3 cos θ=
KII
KI
Note: for positive KI and KII , θ is negative.
27
For KII = 0 (Mode I), θ = 0 for KI = 0 (Mode II), 1− 3 cos θ = 0 ⇒ θ = −70.5◦.
(Note that fθθ is negative for θ = +70.5◦, i.e. hoop stress is compressive, see Fig. 3.5
in Appendix C)
More general solution for KII 6= 0
sin θ
1− 3 cos θ=
KII
KI
⇒ sin θ + 3KII
KIcos θ − KII
KI= 0
Use tan substitution:
sin 2α =2 tan α
1 + tan2 α
cos 2α =1− tan2 α
1 + tan2 α
We get a quadratic in tan (θ/2) i.e.
2t2 − KI
KIIt− 1 = 0
where t = tan (θ/2). There are two solutions to this equation, the maximum hoop stress
is obtained for
tan (θ/2) =14
KI
KII−
√(KI
KII
)2
+ 8
for KII 6= 0
This result is plotted in the earlier figure in terms of M = 2/π(tan−1 KII/K1). Given
KI and KII or given M we can predict the angle of crack growth.
28
1.5 Concept of small scale yielding
All the results so far have been for a linear elastic material. Linear Elastic Fracture
Mechanics (LEFM) can be applied to plastically deforming materials provided the region
of plastic deformation is small. This condition is called the small scale yielding condition.
The concept may be explained by Figs. 1.22–1.26. Here a numerical (finite element)
analysis of a cracked plate is carried out. For simplicity, the plate material is assumed
to be elastic-perfectly plastic (i.e. non-strain hardening). Figure 1.22 shows the finite
element mesh and boundary conditions. Figure 1.23 shows contours of plastic strain
obtained in the elastic-plastic finite element analysis. The upper figure is for plane
strain, the lower is for plane stress. These plastic zones have the characteristic ‘small
scale yielding’ shape in both cases (note the larger plastic zone under plane stress).
Recall from FFM that for plane strain the plastic zone size is approximately given by,
rp =13π
K2
σ2y
and for plane stress the result is
rp =1π
K2
σ2y
,
i.e. three times larger. Here rp is the distance from the crack tip to the plastic zone
boundary and σy is the material yield strength.
Figure 1.24(a) shows the numerically calculated elastic-plastic stress fields directly
ahead of the crack for a plane strain analysis. Here equivalent (Von Mises) stress,
σe, divided by yield strength, σy, is plotted so in the plastic zone σe/σy = 1. The
estimated plane strain plastic zone size is indicated on Fig. 1.24(a). The numerically
calculated value is somewhat less than this approximation. Figure 1.24(b) shows the
same information though a larger scale is used. It is seen that for a large region the
stress fields are well represented by the K field. Figure 1.24(c) shows the same data
again plotted on a log-log plot. Here the zone of K dominance and the plastic zone can
be clearly seen.
Figure 1.25 shows the same results from plane stress and the same trends are seen.
Finally, in Fig. 1.26 a comparison is made between two different specimens, (both
under plane strain conditions) a center cracked plate under uniaxial tension and an
edge cracked plate under bending. (In the figure stress normal to the crack, σyy, is
plotted rather than equivalent stress, σe). The stresses are plotted when both these
29
specimens have the same K value, which is low enough so the plastic zone remains
small. It may be seen that for distances greater than r/a ≈ 0.1 the stress fields in the
two specimens differ but in the K dominant zone, 0.005 < r/a < 0.1, and in the plastic
zone the stress fields are identical. In other words K is controlling the deformation in
the crack tip region and two specimens (e.g. a laboratory specimen and a component)
with the same K value have the same stress and strain fields near the crack. Until
elastic-plastic fracture mechanics was developed, the precise form of these crack fields
was not known—it is not necessary to know them. Provided the small scale yielding
condition holds, two specimens with the same K value have the same crack tip fields.
It is therefore acceptable to work with K and to deem fracture to have occurred when
KI = KIC .
1.5.1 Size requirements for small scale yielding
The requirement for small scale yielding is that the plastic zone size at fracture
should be much less than the crack length. From the equations given earlier it may be
seen that for small scale yielding conditions to hold under plane strain,
rp =13π
K2IC
σ2y
¿ a.
In other words
a À 0.1K2
IC
σ2y
.
The ASTM (American Society for Testing of Materials) specifies that for a valid
plane strain KIC test, the specimen dimensions, crack length a, specimen thickness, B,
uncracked ligament, b, where b = W − a, must be greater than 2.5(KIC/σy)2. This
implies that the specimen dimensions are about 25 times larger than the plastic zone
size. The requirement that the plate thickness, B, is much greater than the plastic zone
also ensures that plane strain rather than plane stress conditions prevail. Under these
conditions, specimens with the same K value will have the same crack tip fields and
fracture will occur when the K value reaches the plane strain fracture toughness value,
KIC .
As will be seen in Section 2, as the specimen size gets smaller or the plastic zone
gets bigger, the small scale yielding condition is not satisfied and elastic-plastic fracture
mechanics must be used.
30
0 10 20 30 40 50 60 70 80 90
0
10
20
30
40
50
60
70
80
90
100
100
100
-1.0 -0.75 -0.5 -0.25 0.0 0.25 0.5 0.75 1.0
0.0
0.25
0.5
0.75
1.0
crack length = 1
Symmetry boundary
Sym
met
ry b
ound
ary
Material behaviour
σy σ
ε
31
Figure 1.23, Crack tip plastic zones for plane stress and plane strain
1
2
3 1
2
3
PEEQ VALUE+0.00E+00
+2.00E-03
+4.00E-03
+6.00E-03
+8.00E-03
+1.00E-02
+4.14E+00
1
2
3 1
2
3
PEEQ VALUE+0.00E+00
+2.00E-03
+4.00E-03
+6.00E-03
+8.00E-03
+1.00E-02
+2.23E+01
Plastic strains
Plastic strains
Plane strain
Plane stress
32
Figure 1.24, Elastic-plastic crack tip fields for plane strain
(a)
(b)
(c)
33
Figure 1.25, Elastic-plastic crack tip fields for plane stress
34
Figure 1.26 Illustration of concept of K dominance for small scale yielding
35
2. Non-linear Fracture Mechanics
LEFM works well as long as the zone of non-linear effects (plasticity) is small compared
to the crack size. However in many situations the influence of crack tip plasticity
becomes important. There are two main issues:
1. In order to obtain KIC in a laboratory test, small specimens are preferred (for cost
and convenience). However, to obtain a valid KIC measurement for materials with
high toughness or low yield strength a very large test specimen may be required
(c.f. size requirements for KIC testing).
2. In real components there may be significant amounts of plasticity, so LEFM is
no longer applicable. For these reasons we need to examine non-linear fracture
mechanics where the inelastic near tip response is accounted for.
2.1 The J integral, (Rice, 1968)
Consider a non-linear elastic material with strain energy density W such that,
σ =∂W
∂ε
(Recall the earlier (equivalent) definition of W , W =∫
σdε.)
Consider a body with an applied stress as shown in Fig. 2.1. Consider an area of the
body A enclosed by the boundary Γ and define the closed line integral, IΓ,
Γ
X
Yσ, ε
n
Figure 2.1 Closed line contour Γ for a loaded body
IΓ =∮
Γ
Wdy − t · ∂u∂x
ds,
36
where t is the traction on Γ, u is the displacement and ds measures distance along the
curve Γ.
2.1.1 Definition of traction, t
The traction vector t on a plane is the average force per unit area exerted by particles
on the positive side of the plane on particles on the negative side of the plane. The
traction vector will depend on the plane considered as illustrated in Fig. 2.2.
t
n
Figure 2.2 Traction t, defined relative to a given plane
The traction is defined as follows:
t = σn
where n is the unit normal to be plane in question and σ is the stress matrix. (This
relationship is known as Cauchy’s theorem and may be stated as follows: The traction
at a fixed point on a surface depends linearly on the normal at the point)
Note that traction is a vector and stress is a matrix (tensor).
2.1.2 Determination of path independent line integral, J
Returning to the integral IΓ.
IΓ =∮
Γ
Wdy − t · ∂u∂x
ds
It can be shown by using the divergence theorem that for an equilibrium stress field
σ and associated strain field ε with W the associated strain energy density, provided
there are no singularities in the region A, the integral IΓ is zero for any path Γ.
This leads us to the definition of the path independent J integral. We now consider
a cracked body and examine the path, Γ shown in Fig. 2.3, where Γ is split into Γ1, Γ2
etc.
37
ΓΓ+
21Γ-
Γ
Figure 2.3 Paths used in definition of the J integral
In Fig. 2.3 Γ2 is the remote boundary, Γ1 surrounds the crack tip, Γ+ and Γ− are
parallel to the top and bottom faces of the crack tip respectively. Since the region
bounded by Γ contains no singularity,
IΓ =∫
Γ2+Γ++Γ1+Γ−
(Wdy − t
∂u∂x
ds
)= 0.
Γ+ and Γ−, are along the crack face and with the axis defined as shown dy = 0. Also,
by definition for a crack, t = 0, i.e. there are no tractions on the crack face.
⇒∫
Γ+
=∫
Γ−= 0 ⇒
∫
Γ2
+∫
Γ1
= 0
or
⇒∫
Γ2
= −∫
Γ1
=∫
Γ1−
where the minus sign for Γ1− indicates that the direction of integration is reversed for
Γ1. We define,
J =∫
Γ
Wdy − t∂u∂x
ds
where Γ is any path starting on the bottom crack face and finishing at the top. The
value of J is constant no matter what path Γ is chosen. (The direction of the integration
is the same as that for Γ2 in the diagram.)
This definition set the stage for non-linear fracture mechanics. The J integral is
path independent for any non-linear elastic material. Plastically deforming materials
can be represented by non-linear elasticity and thus fracture mechanics can be extended
beyond linear elasticity and K. In a similar manner to K, J has an interpretation both
as an energy and a stress quantity. We start with its interpretation as a measure of
stress intensity.
38
2.2 Power law hardening materials—The HRR field
We start with the path independent J integral.
J =∫
Γ
Wdy − t∂u∂x
ds
Take the origin at the crack tip and choose a circular path as shown in Fig. 2.4.
r θ
Figure 2.4 Definition of axes for determination of HRR field
Along this path,y = r sin θ
dy = r cos θdθ
ds = rdθ
,
⇒ J =∫ π
−π
(W cos θ − σ · n∂u
∂x
)r dθ.
Assume a separable solution for displacement, u, as the crack tip is approached,
u = rλ+1u(θ)
ε ∼ du
dr⇒ ε ∼ rλε(θ)
(The symbol ∼ indicates proportional to—we are not interested in the precise form of
the fields at this stage, only in the form of the solution.)
We first wish to determine the value of λ which gives us the order of the singularity
at the crack tip. Since J is independent of path, we can take r as small as desired. If
J is to be finite and non-zero, then we must have
(W cos θ − σ · n∂u
∂x
)→ 1
ras r → 0
Both terms in the bracket are of order O(σε)
⇒ σε → 1r
as r → 0
39
For a linear elastic material, σ ∼ ε. Therefore at the crack tip,
σ ∼ ε ∼ rλ.
⇒ σε ∼ r2λ =1r
⇒ λ = −1/2
i.e. a square root singularity for a linear elastic material.
Now consider a non-linear elastic material with power law hardening (ε ∼ σn ; or
σ ∼ ε1/n ) As before, let
ε ∼ rλε(θ)
⇒ σ ∼ rλn σ(θ)
⇒ σε ∼ rn+1
n λ.
Since
σε → 1r
⇒ λ = − n
n + 1
⇒σ ∼ r−1/n+1σ(θ)
ε ∼ r−n/n+1ε(θ)
This is the HRR singularity, (after Hutchinson, Rice and Rosengren, 1968.
For a more general power law material with uniaxial stress-strain law,
ε
ε0= α
(σ
σ0
)n
,
where ε0, and σ0 are the reference strain and stress and α is a scaling factor, then we
can write,
σij/σ0 = Ar−1/n+1σij(θ; n); εij/αε0 = Anr−n/n+1εij(θ; n),
where A is the undetermined amplitude. By substituting these expression for stress
and strain into the integral expression for J on the previous page, and after some
manipulation, we get,
J = An+1αε0σ0In
40
where In is an integral containing terms which depend only on the hardening exponent
n. We can therefore replace the constant A in the expressions for stress by J above so
σij/σ0 =(
J
αε0σ0Inr
)1/(n+1)
σij(θ; n)
εij/αε0 =(
J
αε0σ0Inr
)n/(n+1)
εij(θ; n).
The terms In, σij(θ; n), εij(θ; n) are dimensionless quantities which depend on
the hardening exponent, n and have been determined numerically. The subscript ij
in the above equations indicate that stress and strain are matrices (tensors) with six
components, i, j = 1 → 3. The distributions for n = 3 and n = 13 are given below. The
intensity of the fields also depends on whether plane stress or plane strain conditions
prevail. Plane strain is always the most severe. This is in contrast to linear elasticity
where the in-plane stresses, σxx and σyy are the same for plane stress and plane strain.
Figure 2.5 Variations of angular stress and strain functions for a Mode I crack under
plane strain. (Hutchinson, J.W., Technical University of Denmark, 1982)
Note that this result also holds for an elastic-plastic power law material, i.e.
ε =
σ
Ewhen σ < σy
εy
(σ
σy
)n
when σ > σy
41
Here α = 1, and εy and σy are used instead of ε0 and σ0 and have the interpretation of
yield strain and yield stress respectively. Close to the crack tip the plastic power law
term will dominate so the HRR field applies.
Thus J fulfills the role of K for a non-linear power law material, J characterises the
intensity of the near tip field. The condition for fracture is simply J = JIC . While power
law hardening is an approximation to the material behaviour, J may still be considered
as a measure of the intensity of the crack tip fields for any material description. In the
same way as K, J will depend on crack length and geometry and also on the magnitude
of the load. There are also tabulated geometry factors for J similar to those for K.
These will be discussed later.
2.3 Crack tip opening displacement
The CTOD approach is commonly used in elastic-plastic fracture mechanics whereby
failure occurs when the crack opening displacement reaches a critical value. For J
dominance the J and the COD approach are equivalent. As we discussed for the K
field, the COD can be determined directly from the crack tip fields, ∆u(r) = 2uy(r, π),
∆uθ
r
Figure 2.6 Definition of crack opening displacement
uy(r, θ) = αεy
(J
αεyσyInr
)n/(n+1)
r uy(θ; n)
Thus, for a given material, (i.e. constant n, α, In, ε0 and σ0) the CTOD depends only
on J and distance r. Since the CTOD depends on r and is zero at r = 0,(except for
perfect plasticity n →∞) the definition of CTOD is somewhat arbitrary.
2.3.1 Conventional definition of CTOD
The CTOD has been defined mathematically as being the crack opening at the
point where 45◦ lines drawn from the crack tip intersect the crack flanks as shown in
Fig. 2.7. It can be shown, using the HRR field, that for this definition of CTOD,
δt = dnJ
σy,
42
where dn depends only on n. For moderate hardening plane stress, dn ∼ 1, for plane
strain, dn ∼ 0.5. Thus there is a one-to-one relationship between CTOD and J for a
given material and any J based approach can be converted to a CTOD based approach.
o
δt
45
45
o
Figure 2.7 Conventional definition of crack opening displacement
The above equation is consistent with the expression introduced in FFM,
δ =G
mσy.
Here the symbol m is used rather than dn and the elastic energy release rate G is used
rather than J . The equivalence between G and J for small scale yielding conditions is
discussed next.
2.4 Relationship between J and G
It can be shown that J is in fact equal to the change in potential energy G for a non-
linear elastic material. The proof is difficult and beyond the scope of the course but an
outline of a proof is given below. Similar arguments to those used in establishing the
relationship between K and G can be employed here. (The original proof due to Rice
did not follow this approach.)
We consider a crack growing by an amount ∆a. Then
Energy released = G∆a ∝ σyy∆u∆a.
For a power law material
σyy ∼ J1
n+1
and
∆u ∼ Jn
n+1
43
so
G∆a ∝ J∆a
and it can be shown that the proportionality constant is equal to 1. The equality of the
line integral J and the energy release rate G holds for any elastic material
J = G = − 1B
∂U
∂a
and for a linear elastic material we have the additional equality
J = G = − 1B
∂U
∂a=
K2
E′ .
While this latter relationship strictly only applies for a linear elastic material, if
the zone of nonlinearity near the crack tip is small, (less than about one twentieth of
the crack length), this latter relationship between J and K may be applied.
As discussed previously, the condition that the plastic zone is small is known as the
small scale yielding condition and is the basis of the use of LEFM for metals. If the zone
of plastic strains is small enough we can ignore J and work with K alone. However,
even under LEFM in ductile metals, the stress and strain fields close to the crack tip
are given by the HRR field, not the K field. Two cracks with the same K value, under
small scale yielding conditions, will have the same J value via the above relation and
therefore the same HRR stress field at the crack tip. Therefore K or J equivalently
characterise the crack field.
The relationship between J and G demonstrates that the J integral may simply
be thought of as another way of obtaining the energy release rate G. However, there
are some difficulties in considering J to be the energy release rate for a real elastic-
plastic material. When a crack grows there is always elastic unloading ahead of the
crack which invalidates the assumptions inherent in the derivation of the relationship
between J and G. However, J still retains its meaning as a stress intensity measure, the
magnitude of the stresses ahead of the crack tip. As stated by Hutchinson: Tempting
though it may be to think of the criterion for initiation of crack growth based on J to
be an extension of Griffith’s energy balance criterion, it is nevertheless incorrect to do
so. That is not to say that an energy balance does not exist, just that it cannot be based
on (the deformation theory) J .†(The meaning of the term deformation plasticity here
† J.W. Hutchinson, Journal of Applied Mechanics, 1983 (see Appendix A).
44
is equivalent to the term non-linear elasticity.) Under large scale yielding conditions
which we will discuss later, J is sometimes split up into elastic and plastic parts, i.e.
Jtotal = Je + Jp
The term Je is given by
Je =K2
E′
i.e. it is the value of J if there was no plasticity. Sometimes (and somewhat confusingly
in view of its use as a symbol for the non-linear elastic energy release rate) the symbol
G is used to indicate the elastic part of J .
2.5 Evaluating J for test specimens and components
We have seen the importance of J in non linear fracture mechanics. The next question
is how to evaluate it. The line integral approach is rather awkward and usually requires
numerical techniques such as finite element analysis, so approximate methods have been
developed to estimate J in test specimens and in actual components.
In this section we will discuss two of the most popular methods estimate J—the
use of the η factor and the GE-EPRI J estimation methods for power law materials.
Most other J-estimation procedures are based on these two methods.
To evaluate J using the η factor we exploit the relationship between J and G. To
show how this is done, it is helpful to revisit the concept of the limit load or plastic
collapse load of a specimen.
2.5.1 Limit load and the definition of η
We examine an elastic-perfectly plastic material. The limit load (sometimes called the
collapse load) is the load at which plastic collapse occurs for such a material. Consider
a beam in bending (Fig. 2.8).σ
σB − y
y
WML =
σyBW 2
4
Figure 2.8, Illustration of limit moment for a plastic beam in bending.
Next consider a cracked beam in bending with a << W .
45
M
a
WM
a
B
Figure 2.9, Edge cracked beam in bending.
It can be shown that for plane stress conditions,
MLC = σyB(W − a)2
4= σy
W 2B
4(1− a/W )2
Note that W is replaced by W −a, i.e. the collapse moment for the cracked plate is the
same as that for a plate of width W − a. The additional subscript ‘C’ here emphasises
that it is the solution for a cracked plate. Often the ‘C’ is left out.
For a center cracked plate with crack length 2a and plate width 2W , in tension with
a << W , subjected to a load 2P , the limit load under plane stress conditions, is given
by
2W
PLC = σy(W − a)B.
Limit load solutions are commonly used in fracture mechan-
ics. The ratio between load and limit load is a measure of the
extent of plasticity and provides a good means of compar-
ing two geometries. For example two different geometries
at the same ratio of load to limit load have similar J in-
tegral values. Limit loads have been obtained analytically
and numerically and have been tabulated for a wide range
of geometries;
For a three point bend geometry of length 2L under plane strain conditions and using
the Von Mises yield criterion:
PLC = σy2√31.22
(1− a
W
)2 W 2B
2L.
46
This solution may be compared with that for a small crack under pure bending given
earlier. For a three point bend bar of length 2L with applied load P , the moment at
the crack plane is M = PL/2, and the 1.22 term in the equation is due to a finite sized
crack.
2.5.1.1 Plane strain limit load
The 2/√
3 term in the limit load definition above is due to the plane strain condition,
i.e. under plane strain uniaxial tension, σ
σyy = σ, σxx = 0, σzz = 0.5(σxx + σyy) (in plasticity)
then for von Mises yield,
2σ2vm = (σxx − σyy)2 + (σxx − σzz)2 + (σyy − σzz)2
⇒ σvm =√
32
σ
At yield σvm = σy, therefore for yielding under tension in plane strain σ = 2σy/√
3, so
the limit load is higher by a factor of 2/√
3 under plane strain conditions compared to
plane stress conditions when the yield condition is simply σ = σy.
2.5.2 Definition of reference stress
A quantity closely related to the limit load is the reference stress. The reference stress
is a measure of the proximity to collapse of the structure, and is defined as:
σref = σy
(P
PL
).
Thus, plastic collapse will occur for a perfectly plastic material with yield strength
σy when σref = σy and since PL is proportional to yield stress, σref is independent of
σy. The reference stress is a correction to the applied stress to take account of the
effect of the crack on the response of material. Use is made of the reference stress in
the application of structural integrity assessments, as will be seen later in the course.
2.5.3 Use of limit load to define the η factor
Limit loads are useful in determining the η parameter, which is used to relate J to the
area under the load displacement curve. Recall the energy definition of J ,
J = − 1B
∂U
∂a.
47
Under applied displacement,
U = Ue =∫ ∆
0
P (∆) d∆ ⇒ J = − 1B
∫ ∆
0
∂P
∂ad∆.
It can be shown that this is equivalent to writing
J =η
B(W − a)
∫ ∆
0
P d∆ =η
B(W − a)A,
where η is a geometry factor related to the compliance and A is the area under the load
displacement curve. The above has been derived for applied displacement. However
since J is independent of the mode of loading it also holds for applied load.
This form for J is convenient because, provided η is known, J in an experiment
can be obtained from a load displacement history, which is easy to measure.
Consider a three point bend specimen with a ¿ W under plane stress made of a
rigid elastic, perfectly plastic material.
PL = σyW 2B
2L(1− a/W )2
Figure 2.11, Limit load for an edge cracked beam under three point bending.
Under displacement control the potential energy, U , is given by, U = Ue = P∆ (see
Fig. 2.12). Therefore
U = P∆ = Pl∆ = σyW 2B
2L(1− a/W )2∆.
P
∆
∆
L
P
U = P
Figure 2.12, Load-displacement behaviour for a rigid-elastic, perfectly plastic material.
48
We can differentiate the above expression for U , keeping displacement fixed, to obtain
J ,
J = − 1B
∂U
∂a
∣∣∣∣∆
=2W
σyW 2
2L(1− a/W )∆.
Since P = PL = σyW 2B
2L(1− a/W )2 ⇒ W 2
2L(1− a/W ) =
PL
B(1− a/W )
⇒ J =2
B(W − a)A.
i.e. η = 2 for this load configuration.
For a center-cracked-tension geometry,
, ∆/2
W
, ∆/2 U = P∆ = σyB(W − a)∆
J = − 1B
∂U
∂a
∣∣∣∣∆
= σy∆ =1
B(W − a)P∆
i.e. η = 1 for this loading configuration.
Note that if, as is often the case, the crack
length is designated 2a instead of a, then in
the J equation the load per crack tip, i.e. P/2,
must be used to have η = 1.
The above is an illustration for perfect
plasticity. There are more rigorous proofs,
(given in Kanninen and Poplar) which show
that in general for a low hardening material,
η is close to 1 in tension and 2 in bending.
Many crack geometries are loaded by a combination of bending and tension. e.g.
for a compact tension specimen,
η = 2 + 0.52(1− a/W ).
2.5.4 η value for a linear elastic material
We can also evaluate η for a linear elastic material.
49
Figure 2.14, Load-displacement curve for a linear elastic material
For a linear elastic material:
J = G =1
2BP 2 dC(a)
da,
where C is the elastic compliance
P =∆C⇒ P 2 = P
∆C
J =1
2BP
∆C
dC
da=
1B
1C
dC
daA
and since the alternative equation for J is
J =η
B(W − a)A
⇒ ηe =W − a
C
dC
da
Thus if the compliance is known, ηe can be determined. The subscript ‘e’ is used here
to emphasise that this is the value for an elastic material. In general ηe and ηp are not
equal even for the same geometry.
2.5.5 Evaluating J for an elastic-plastic material
Most materials are elastic-plastic and when a specimen is loaded part of it will have
yielded while the rest will be elastic. Under these conditions the total J value may be
estimated by
J = Je + Jp ≈ ηe
B(W − a)Ae +
ηp
B(W − a)Ap
50
where Ae = 1/2(P∆e) and Ap is the remaining portion of the load displacement curve
as shown in Fig. 2.15.
Figure 2.15, Load displacement curve for an elastic-plastic material
Alternatively, are more usually, the elastic part of J may be determined from the
linear elastic K value, which is easily obtained, and then
J =K2
E′ +ηp
B(W − a)Ap
The J value in an experiment can then be obtained simply by measuring the area
under the load-displacement curve. For a deeply cracked bend specimen, ηe ≈ 2, so a
commonly used J approximation for a deep cracked bend specimen is,
J =2
B(W − a)A,
where A is the total area under the load displacement curve.
2.5.6 GE-EPRI J Estimation Scheme
The GE-EPRI scheme is an alternative approach to calculating J . GE-EPRI stands for
General Electric-Electrical Power Research Institute, where the method was developed.
Consider a body loaded by remote stress σ∞ shown in Fig. 2.16.
51
σ∞
σ, ε�����
σ∞
Figure 2.16 Cracked body under remote tension
First consider a linear elastic material with,ε
ε0=
σ
σ0(E = σ0/ε0)
where σ0 and ε0 are a normalising stress and strain (material parameters).
By dimensional analysis we can write
σ
σ0(x, y) = [σ∞/σ0] f(x, y)
ε
ε0(x, y) = [σ∞/σ0] g(x, y)
where σ and ε are the stress and strain at any point in the body (Fig. 2.16). This
simply states that stress and strain in a linear elastic body are proportional to applied
load.
Since
J =∫
Γ
Wdy − t∂u∂x
ds
⇒ J ∝ (σ∞/σ0)2σ0ε0L
where L is an appropriate length. Note that because we integrate with respect to
distance along the contour Γ the dependence on x and y does not enter the expression
for J . We can write
J = aσ0ε0
[σ∞σ0
]2
H(a/W )
52
Here the characteristic length has been taken to be the crack length a and H is the
proportionality constant which depends only on geometry. Compare with
K = σ∞√
aY (a/W ).
Since for a linear elastic material, J = K2/E′
J =σ2∞aY 2(a/W )
E(in plane stress)
=σ2∞ε0aY 2(a/W )
σ0
=[σ∞σ0
]2
σ0ε0aY 2(a/W ).
So for linear elasticity, H(a/W ) = Y 2(a/W ). This is (yet another) way of expressing
J (or K) for a geometry. This particular normalization proves useful as it can be
generalised to different types of geometries and materials.
For a plastic (or non-linear elastic) material with power law hardening behaviour,
ε/ε0 = α(σ/σ0)n
σ
σ0(x, y) = [σ∞/σ0]f(x, y, n)
ε
αε0(x, y) = [σ∞/σ0]ng(x, y, n)
Note the additional dependence on the hardening exponent n. Using the same
argument as before we can write
J ∝ ασ0ε0(σ∞/σ0)n+1L
and thus
J = ασ0ε0a
[σ∞σ0
]n+1
H(a/W, n).
It proves useful to normalise by the limit load rather than σ0 so we rewrite as
J = ασ0ε0a
[P
P0
]n+1
h(a/W, n)
where P is remote load and P0 is limit load. The function h depends only on n and
a/W and can be tabulated in a similar fashion to Y . Note the dimensions of each term
53
in the expression:J [stress][length]α dimensionlessσ0 [stress]ε0 dimensionlessa [length]
P/P0 dimensionlessh dimensionless
Some values of h are shown in Fig. 2.17 for an edge cracked panel subjected to
bending for a range of a/W ratios and strain hardening exponent n.
a/W=1/8
Figure 2.17, h function for edge cracked panel in bending.
Note that except for the shallow crack, (a/W = 1/8) at high n, h is quite weakly
dependent on n and close to unity. Typical values of n for metals range between 5 and
20 (0.05 < 1/n < 0.2)
2.5.7 Elastic-plastic material behaviour
The function h is based on purely plastic (power law) behaviour. For elastic-plastic
behaviour with
ε =
σ
Ewhen σ < σy
εy
(σ
σy
)n
when σ > σy,
J can be partitioned as before into elastic and plastic parts,
J = Je + Jp
54
with Jp evaluated using the estimation scheme above (taking α = 1, ε0 = εy and σ0 = σy
and
Je = K2/E′.
In order to get good agreement with numerically calculated J values in the elastic-
plastic regime, Je is usually adjusted slightly. The precise form of Je used in the EPRI
scheme is described in Section 5.4 of the book by Kanninen and Poplar. It is based on
a plastic zone correction approximation for J as discussed below.
2.5.7.1 Plastic zone correction
When the plastic zone size is relatively small (on the order of a tenth of the crack length)
modifications to the LEFM K are sufficient to account for the effects of material non-
linearity. The effect of crack tip yielding is to reduce the effective load supporting area
at the crack tip relative to an equivalent linear elastic material. This effect may be
accounted for approximately by using an effective crack length, ae, in the definition for
K, i.e.,
Keff = Y (ae)σ√
ae
where Y is the LEFM geometry factor. The effective crack length ae is given by,
ae = a + ry,
where a is the actual crack length and ry is half the plane strain or plane stress plastic
zone size.
Consider the case of a small crack in a large plate under tension and use the plane
stress expression for the plastic zone size,
ry =12π
(K
σy
)2
.
For a small crack, Y =√
π (i.e. independent of crack length) and we get
Keff =√
πσ√
a√1− 1
2
(σ
σy
)2.
Using the plane stress plastic zone size ensures that we will get the highest value of
Keff , Keff will be overestimated if conditions are predominantly plane strain.
2.5.8 Overall J estimation procedure
55
The final form of the GE-EPRI J estimation scheme, is then
J = Je(aeff) + Jp(a, n),
with Jp evaluated using the estimation scheme and Je evaluated using the plastic zone
correction as discussed above, with ry based on the unmodified crack length, a. A com-
parison between the numerically calculated J value and the GE-EPRI approximation
is shown in Fig. 2.18 (This figure is adapted from the figure on page 318 of Kanninen
and Poplar). The GE-EPRI scheme can be used to estimate J in materials which obey
power law hardening in the plastic regime. However, in order for the estimate of J to be
accurate, the material behaviour should be well approximated by a linear elastic-power
law hardening law.
Solid line: finite element solution Dashed line: GE-EPRI approximation
Figure 2.18, Comparison between GE-EPRI J estimation scheme and finite element
calculations for n = 3 and n = 10.
Note that the η approach and the GE-EPRI scheme are equivalent in practice. The
values of η and h must be determined numerically, though good approximations can
often be made, e.g. taking η = 2 for bending, 1 for tension. Exact values of the J integral
could be obtained from a full 3-D finite element analysis of the specimen/component
and calculating J using the line integral definition. However, this is expensive on time
and resources so approximate schemes are preferred.
Note also that occasionally the symbol G is used to indicate the linear elastic energy
release rate which is equal to the elastic J value, Je. This should not be confused with
56
the non-linear elastic energy release rate, also given by the symbol, G or here G and is
equal to the total J .
We have shown how to calculate J for test specimens. However, to carry out a
failure assessment, we still need to determine J in the actual component we are inter-
ested in. Again this must be done either numerically or by some other approximation
technique. Failure assessment procedures such as BSPD 7910, to be discussed in more
detail later incorporate methods to do this and in the elastic-plastic fracture regime
they are simply approximation schemes for the J integral.
2.6 Application of non-linear fracture mechanics
The theoretical basis behind the application of non-linear fracture mechanics is
illustrated by the figures overleaf. Figure 2.19 shows numerically calculated elastic-
plastic stress fields in the vicinity of a crack at different load levels for two different
geometries—a three point bend geometry (tpb) and a centre crack panel in tension
(ccp). The tpb geometry has a plate width, W , of 40 mm and and a crack length to
specimen width, a/W , of 0.5. The ccp geometry has W = 400 mm (i.e. a very large
plate) and a/W = 0.1. In each figure the K value for the tpb and ccp specimen are the
same. Here the material behaviour in the elastic-plastic regime is assumed to follow a
power law relation with yield stress, σy = 500 MPa and n = 5.
If both specimens are loaded at a low load, K = 5 MPa√
m, then it is seen that
the specimens deform predominantly elastically and the crack tip stress distributions
are the same.
As the load is increased to K = 30 MPa√
m the region of plastic deformation
increases but as ‘small scale yielding’ conditions are satisfied both specimens still have
the same stress distribution at the same K value. Note, however, that the near tip fields
are represented by the HRR field rather than the K field.
At the highest load level, (K = 85 MPa√
m) the zone of K dominance has almost
disappeared for the smaller tpb specimen. However, though not shown, the near tip
fields are still represented well by the HRR distribution. It may also be seen that the
crack tip stresses are different for both specimens even though the elastic K value is the
same. (The difference is not very obvious on the log-log scale.)
The reason for this difference is explained by the J versus K plot shown in Figure
2.20. It is seen in Fig. 2.20(a) that, because the ccp specimen is larger, the small scale
yielding condition holds up to the maximum K value applied (i.e J = K2/E′ holds for
this specimen up to 85 MPa√
m). For the tpb specimen, however, as the plastic zone
57
size increases, the J value exceeds that given by the small scale yielding estimate.
Figure 2.20(b) illustrates that, as expected, under small scale yielding conditions
both specimens at the same K level have the same near tip stress distributions. However,
at the largest load level, as seen from the J vs. K diagram, the J values for the tpb
specimen is higher than that for the ccp specimen. Hence, the tpb specimen has a
somewhat higher stress field than the ccp specimen. Under these conditions the stress
fields are not characterised by K and J (and the HRR field) must be used instead.
Figure 2.19, Comparison of K field and numerical crack tip fields for two geometries
40 mm 20 mm
800 mm
80 mm
tpb
ccp
58
Figure 2.20, (a) J plotted against K for ccp and tpb geometry, (b) comparison between K field and numerical stress field at K = 30 MPa m1/2 (c) comparison between HRR field and numerical stress field at K = 85 MPa m 1/2
K = 85 MPa m1/2
(a)
(b)
(c)
59
2.7 K dominance, J dominance and size requirements
As discussed earlier, far away from the crack tip the K solution is invalid. Close to the
crack tip there will be non-linearity (plasticity) so the K-field will not be valid there
either. In a KIC test, there must be a region where the K field is in agreement with the
stress fields in the specimen. Assuming we have sufficent thickness to guarantee plane
strain conditions, in order to have a valid KIC test, we require a small plastic zone, rp.
For Mode I,
rp =13π
(K
σy
)2
= 0.1(
K
σy
)2
.
We require a À rp. The ASTM requirements are
a, B, W/2 > 2.5(
KIC
σy
)2
i.e. specimen dimensions are about 25 times larger than the plastic zone size, e.g. For
AISI 4340 steel, KIC = 65 MPa√
m, σy = 1400MPa then minimum plate size W is
11 mm. But for A533B nuclear reactor grade steel, KIC = 180MPa√
m, σy = 350MPa,
minimum W for a valid KIC test is 1.3 m.
For JIC testing size requirements are less stringent. The requirement is that the
J solution (i.e. the HRR field) dominates over a region significantly greater than the
crack tip opening displacement. (In this region the stresses are strongly affected by the
blunting of the crack.) It has been found from numerical studies that bend geometries
show a larger zone of J dominance than tension geometries. Therefore different size
requirements are needed for these two types of geometries.
For a centre crack tension specimen the zone of J dominance becomes vanishingly
small relative to the crack tip opening displacement when J > (6× 10−3)aσ0. In other
words the size requirement for a tension specimen is that
a > 150JIC/σy.
For a deeply cracked bend specimen, J dominance is maintained up to about J =
0.07aσy giving a size requirement that
a > 15JIC/σy.
In practice the ASTM JIC standard recommends that deeply cracked bend
specimens—the compact tension specimen (which despite its name is a bend dominated
60
geometry) or the single edge bend (three point bend) specimen, are used to obtain JIC
and that
a, b, B > 25JIC/σy.
These limits have been confirmed from experiments, i.e. as long as these requirements
are met the JIC value obtained is independent of the specimen size or type—it is a
material property.
Ii should be pointed out that since both KIC and JIC are material properties, KIC
can always be calculated from JIC using the relationship
KIC =√
JICE′.
So a valid JIC test can be used to obtain KIC . The ASTM standard E1820 designates
a KIC calculated from a JIC as KJIC , though in principal, KJIC ≡ KIC
The concept of J dominance is illustrated by Figure 2.21. (The geometries analysed
are the same as those shown in Figs. 2.19 and 2.20, though the loads are higher). Figure
2.21(a) and (b) illustrate the stress fields for a tpb specimen at two different J levels
(note the way the x axis is normalised). It may be seen that at J/aσy = 0.004 the
HRR field gives a good representation of the stress fields in the specimen for distances
on the order of the CTOD (CTOD ≈ 0.5J/σy so the x-axis extends for about 10 crack
tip openings.). At the higher load, Fig. 2.21(b), J dominance is lost and the HRR field
agrees with the stress fields only very close to the crack tip, at distances less than the
COD.
For the ccp geometry, as shown in Fig. 2.21(c), even at the lower normalised J
value the HRR field is not close to the stress fields in the specimen. The J levels at
which J dominance for these two specimens are lost are consistent with the ASTM
limits specified earlier.
It should be pointed out that when J dominance is lost the stress field will always
fall below the HRR distribution. In other words it would be conservative to assume that
the fields are J controlled. However, when determining the fracture toughness, it is of
course important to test the worst case situation, which will be the J dominance case.
This is why the standards specify the use of a bend geometry with size requirements to
ensure J dominance. In recent years, attention has turned to taking advantage of some
of the conservatism inherent in the applicant of J based fracture mechanics to cracks
loaded predominantly in tension and with significant amounts of plasticity. However,
this work is still at the research stage. Note that the loss of J dominance does not imply
61
that J loses its path independence. It simply means that the crack tip fields are not
well described by the HRR field with amplitude J .
Figure 2.21, Comparison of numerical stress fields and HRR fields for (a) and (b) tpb geometry and (c) ccp geometry
(a)
(b)
(c)
The idea of K and J dominance is summed up by Fig. 2.22, which shows schemat-
62
ically the K and J dominance zones at two different loads. The process zone, indicated
in the figure, is the damage zone near the crack tip, whose size is on the order of the
CTOD. In order for J dominance to hold, the region where the HRR fields agree with
the stress fields in the specimen, as determined by a finite element analysis for example,
must be larger than this process zone size.
HRRfield
ProcessZone
Specimen
K field
HRRfield
ProcessZone
Specimen
ILLUSTRATION OF K DOMINANCE (at low load, or very large specimen)
ILLUSTRATION OF J DOMINANCE(at higher load or small specimen)
Figure 2.22, Illustration of K and J dominance in elastic-plastic specimens.
63
2.8 Standard test to determine JIC
In this section we discuss the ASTM E 1820–01 test standard for measurement of frac-
ture toughness. A three point bend or compact tension specimen is recommended with
0.45 < a/W < 0.7 and a valid test requires that
a, b, B > 25JIC
σy.
The load and load point displacement of the specimen are measured during the test
and J is calculated from the area under the load displacement curve. JIC is the value
of J at crack initiation.
For very ductile materials an obvious initiation toughness, JIC is difficult to mea-
sure as there is usually some stable ductile tearing before final failure. For such mate-
rials a J resistance curve, (J versus ∆a) is measured and the curve extrapolated back
to ∆a = 0 to obtain JIC . The change in crack length ∆a may be obtained using the
unloading compliance method (see section 2.8.1) or some other method. J is then plot-
ted versus ∆a and JIC is determined by extrapolating back to ∆a = 0. Crack blunting
can give an apparent ∆a even though crack growth has not occurred and this must be
accounted for in determining JIC . This is discussed in more detail in ASTM E 1820.
2.8.1 Compliance method to estimate ∆a
Crack growth is generally non-uniform through the thickness of the specimen. As
the centre of the specimen is under plane strain conditions crack growth tends to be
higher there. Therefore inspection methods which rely on observing the crack growth
on the surface of the specimen may not be sufficiently accurate. Furthermore, such
inspection requires some operator judgement and are difficult to automate. The crack
compliance method avoids both of these problems (to some extent).
An elastic-plastic material unloads elastically and we have the relationship
∆ = C(a)P.
If a specimen is unloaded a small amount during testing the elastic compliance can be
obtained from the load-displacement curve (see Fig. 2.23). If the compliance changes
this can only be due to a change in crack length. The function C(a) is tabulated for
standard fracture specimens, and thus if the compliance is measured the amount of crack
growth ∆a can be inferred. If the crack growth is non-uniform through the thickness,
then this crack length will be the average crack length through the specimen.
64
Figure 2.23, Estimation of crack length using linear elastic compliance
An alternative approach in constructing a J − ∆a resistance curve is to stop the
test, break open the specimen and determine the amount of crack growth optically. The
disadvantage of this method is that multiple specimens will be needed to construct the
J-resistance curve.
Even if the compliance method is used to determine the crack length, the specimen
should be broken open at the end of the test to compare the actual initial and final
crack length with the values estimated using the crack compliance method.
65
3. Micromechanisms for ductile and brittle fracture
The student is referred to the extracts from the paper by Ritchie and Thompson, “On
macroscopic and microscopic analyses for crack initiation and crack growth toughness
in ductile alloys” (see Appendix A).
As discussed in previous sections, the criterion for a material to fail by fracture is
that, J = JIC . Provided J can be obtained and JIC is known (or can be measured)
we can determine whether or not fracture will occur. However, this does not tell us
anything about the mechanism of fracture, for example, whether fracture will be by a
brittle or ductile mode. We have already looked at a simple cleavage model for brittle
failure. The model for cleavage is the RKR (Ritchie, Knott, Rice) model which says
that ‘brittle crack extension occurs when the local tensile opening stress ahead of the
crack exceeds a local fracture stress over a microstructurally significant distance.’
3.1 Micromechanism of cleavage failure
Recall from first year mechanics of materials that the ideal strength of a crystal is on
the order of E/10, which should be the stress required for cleavage. However, due
to yielding and crack blunting, classical plasticity, predicts that these stress levels will
never be reached even at the tip of a crack—the maximum stress at the tip of a blunting
crack is about 3σy. In order to explain why cleavage fracture can occur an additional
micro-mechanism is required. The generally agreed micro-mechanism for steels is that
the dislocations that are emitted from the crack tip build up at the adjacent grain
boundary, amplifying the local stress (see Fig. 3.1).
Dislocation pile-up
Carbide particles
Figure 3.1 Schematic of cleavage failure at the microscale
The stress at the head of the dislocation pileup (at the grain boundary) is n times the
stress at the crack tip, where n is the number of dislocations. This stress may then
be large enough to initiate failure at grain boundary inclusions (e.g. carbides for a
ferritic steel) and failure of the inclusion triggers failure in the associated ferrite grain
66
(in a carbon steel) The resultant micro-crack then links up with the main crack and the
crack grows unstably.
Using dislocation arguments, the stress required for cleavage, σ∗f , is given by
σ∗f ≈Gγm
5√
dg
,
where G is the shear modulus, γm is the relevant surface energy (including local plastic
work) and dg is the grain size. Since G and γm are weakly dependent on temperature,
this cleavage stress σ∗f is relatively independent of temperature—for a typical carbon
steel it is about 800 MPa.
3.2 Prediction of fracture toughness using the RKR model and the HRR
field
Using the RKR model and the HRR field we can write down a micro-mechanics based
failure equation (see Fig. 3.2). The RKR criterion is that for a Mode I crack, σ22 = σ∗f ,
at r = rc where, from our micro-mechanical model above, rc is on the order of a grain
size. (Following the notation of Ritchie and Thompson, l∗0 is used here instead of rc).
σσ αε σ
σ22
0 0 0
1 1
220=
+
J
I rn
n
n/
~ ( ; )
σ22
σ22
Figure 3.2 RKR model for cleavage failure
The failure equation becomes,
σ∗fσ0
=(
JIC
αε0σ0Inl∗0
)1/n+1
σ22(0; n).
Now, ε0 = σ0/E, and α and In are fixed for a given material and temperature so we
rewrite:
σ∗f = Aσ0
(EJIC
σ20l∗0
)1/n+1
,
where
A = σ22(0; n)(
1αIn
)1/n+1
67
⇒ EJIC =(
σ∗fAσ0
)n+1
σ20l∗0.
As discussed previously, there is a one-to-one relationship between JIC and KIC ,
i.e. JIC = K2IC/E′. Then
KIC =√
E′JIC =(
σ∗fAσ0
)(n+1)/2
σ0
√l∗0,
(Taking E′ = E for simplicity). Rearranging, we get
KIC =1
A(n+1)/2(σ∗f )(n+1)/2σ
(1−n)/20
√l∗0.
For n = 10 (a typical value for steel), the values are, In = 4.54, σ22 = 2.5, ⇒A ≈ 2.2 (taking α = 1). It may be assumed that A is independent of temperature (n
depends weakly on temperature). Inserting these values gives,
KIC = 0.01(σ∗f )5.5σ−4.50
√l∗0.
A direct relationship between KIC , the failure stress, the yield strength and the grain
size is obtained (since in this model, l∗0 = dg). Consider the effect of an increase in
temperature on the fracture toughness. If temperature is increased, σ∗f is unchanged,
σ0 decreases, so KIC increases. If the grain-size is decreased both σ0 and σ∗f increase
However, the relative increase in σ∗f is higher, so the overall effect is to increase KIC .
3.3 Micromechanism of ductile failure
At high temperatures the yield stress decreases so the crack tip stresses go down and
cleavage fracture becomes less likely. The dominant failure mechanism is then ductile
tearing—a process known as micro-void coalescence. In steels, voids are initiated by
debonding from large inclusions, e.g. manganese sulphide particles. These voids grow,
coalesce and link up with the main crack, leading to slow and stable tearing with a large
amount of absorbed energy (see Fig. 3.3).
Sulphide particles (Inclusions)
*0l
Figure 3.3 Micromechanism of ductile failure
68
Void growth is generally said to be a strain-controlled phenomenon, as opposed
to cleavage which is stress controlled. Ductile fracture occurs, when the plastic strain
reaches the critical plastic strain, ε∗f at the critical distance, r = rc. In this case the
critical distance is taken to be the spacing between the voids (or void initiating particles),
see Fig. 3.3.
3.4 Prediction of fracture toughness using the MVC model and the HRR
field
Assuming that the crack tip strain is represented by the HRR field and that failure
occurs in the plane of the crack (θ = 0) the failure equation is,
ε∗fε0
=(
JIC
αε0σ0Inl∗0
)n/n+1
ε(0; n).
(Note that there is a typographical error in the Ritchie and Thompson paper.) The
quantity ε is the equivalent (Von Mises) plastic strain as given by the HRR distribution.
For large n, n/(n + 1) ≈ 1 so we can write the above equation as,
JIC =ε∗fε
(ασ0Inl∗0).
Writing in terms of KIC we get,
KIC =
√αIn
ε
√σ0E′Inε∗f l∗0.
This equation predicts that increasing temperature will decrease the ductile fracture
toughness (since σ0 decreases with increasing temperature).
3.4.1 Definition of critical plastic strain
The question arises as to what value of strain to use in the equation above, i.e. will
it simply be the failure strain measured in a tensile test? Considerable research on
micro-mechanics models for void nucleation and growth has been carried out, see e.g.
the textbook by Webster and Ainsworth. It has been shown that the rate of void growth
rate is proportional to the hydrostatic stress and the plastic strain rate, i.e. void growth
actually depends on stress as well as strain.
A typical void growth model is that of Rice and Tracey who found that if the
material containing the void is assumed to be elastic-perfectly plastic with yield strength,
σy, then the void growth rate is given by
r = 0.558r sinh (1.5σ∞m /σy)ε∞p ,
69
where r is the void radius, r is the rate of increase of the radius, σ∞m is the remote mean
(hydrostatic) stress and ε∞p is the remote plastic strain rate. (The notation ∞ is used
here to indicate that the stress and strain fields are those remote from the void, but
when used to link with fracture mechanics analyses, these will be the crack tip stress
and strain, i.e. the size scale of the void is understood to be considerably smaller than
the size scale of the crack tip singularity)
Since the void growth rate depends on the hydrostatic stress, and failure is associ-
ated with the growth and coalescence of voids, this means that the value of the plastic
strain, ε∞p , at failure will depend on σm/σy. Therefore ε∗f cannot be determined directly
from a tensile test since the stress state (σm/σy) is different. The ratio σm/σy or σm/σe,
where σe is the equivalent stress, is generally known as the stress triaxiality.
A direct equation for the failure strain is obtained by integrating Rice and Tracey’s
model and assuming that failure occurs when the void reaches a certain size (e.g. large
enough to link with neighbouring voids (see Fig. 3.4)).
dp
Dp
initial crack
(a)
(b)
Figure 3.4 (a) Initial configuration of voids in a ductile material (not to scale) (b) Void
coalesence leading to ductile crack growth.
Given the Rice and Tracey equation,
r = 0.558r sinh (1.5σ∞m /σy)ε∞p ,
we can integrate from the initial void size (or void initiating particle), Dp, to the fi-
nal void size,which is given by the initial spacing, dp (see Fig. 3.4). Assuming that
triaxiality, σm/σy, is constant during void growth, we can integrate the above equation,
dr
r= 0.558 sinh (1.5σm/σy)dεp
⇒∫ dp/2
Dp/2
dr
r= 0.558 sinh (1.5σm/σy)ε∗f ,
70
where ε∗f is the failure strain corresponding to the particular triaxiality.
Carrying out the integration, we get
ε∗f =ln (dp/Dp)
0.558 sinh (1.5σm/σy).
Generally, to allow for strain hardening, σy in the above equation is replaced by the
equivalent Mises stress σe. An alternative relationship which is only accurate for high
triaxiality conditions (σm/σe > 0.8) is
ε∗f =ln (dp/Dp)
0.283 exp (1.5σm/σe).
We can eliminate the term ln (dp/Dp) by considering the case of a uniaxial test.
For this geometry, σe = σ, where σ is the applied stress and σm = σ3, so σm/σe = 1/3.
If failure occurs in a uniaxial tension test when strain = εf , then inserting this into the
above equation and eliminating the ln term, we get
ε∗f =0.521εf
sinh (1.5σm/σe).
(Note that it is not correct to use the high triaxiality equation to obtain a similar
relationship to the one above, though it is used in a number of text books.)
The above relationship gives an expression for the critical strain, ε∗f in terms of the
uniaxial failure strain, εf which can be easily measured.
3.4.2 Use of notched specimens to study triaxiality effects
Notched specimens can also be used to examine experimentallythe effect of stress
state (triaxiality) on failure strain , see Fig.3.5
amin
Notch radius, ρ
Figure 3.5 Notched specimen used to determine effect of stress state on failure strain.
71
By varying amin and ρ the triaxiality, σm/σ, in the notch region, is varied. The Bridge-
man equation gives an approximate solution for the triaxiality in the notch as
σm
σe=
13
+ ln(
1 +amin
2ρ
).
If we then measure the strain in the notch at failure we can generate a plot of failure
strain, ε∗f , versus triaxiality as shown schematically in Fig. 3.6.
0.3
3.0 sharp crack
0.2 1.2
σσ
m
εf
uniaxia l
*
Figure 3.6 Effect of triaxiality on failure strain as determined from notched bar tests
The critical strain for a crack can be determined by extrapolation. Note that
the dependence of failure strain on triaxiality will depend on the material. For some
materials the Rice-Tracey expression works well but for others it may not work so well.
3.5 Competition between brittle and ductile fracture
Based on our two relations for cleavage and ductile failure, we can now construct two
failure curves as shown in Fig. 3.7. In general these mechanisms will be in competition
and fracture occurs by the mechanism that is satisfied first. At low temperatures it is
seen that the cleavage criterion will be satisfied before the ductile tearing condition and
at high temperatures the ductile mechanism is activated first.
We can define a transition temperature at which the failure mechanism changes
from a cleavage to a ductile mechanism. In reality this transition does not occur at
a single temperature— there is a gradual change from predominantly cleavage failure
to predominantly ductile as the temperature is increased. The high toughness regime
where failure is by ductile tearing is often called the upper shelf regime.
The mode of failure can often be identified by examination of the broken surfaces
of the fractured specimens. The fracture surface of a material which has failed in a
72
ductile fashion tends to be rough and dimpled indicative of the void growth mechanism
while the fracture surface of a material which has failed by cleavage tends to be flat and
shiny (see Fig. 3.8).
( ) ( )KA
lIC n f
nn=
+
+−1
1 2
1 2
0
1 2
0/
*( )/
( )/ *σ σ
KIC
6GORGTCVWTG
$TKVVNG 6TCPUKVK
���VGORGTCVWTG
&WEVKNG
KI
E lIC
n
f= ′
αε
σ ε~* *
0 0
Figure 3.7 Competition between brittle and ductile failure.
(a) (b)
Figure 3.8 Typical fracture surfaces in metals (a) cleavage (b) microvoid coalescence.
Further discussion on this topic is provided in the paper by Ritchie and Thompson in
Appendix A.
73
4. Application of BS 7910 in failure assessments
BS7910 is the current British standard which is used in the assessment of flawed struc-
tures. Its full title is “Guide on methods for assessing the acceptability of flaws in
metallic structures”. The purpose of the standard is to provide a simple, repeatable
procedure for assessing the safety of cracks. It also makes contact with other British
and International Standards (e.g. ISO, ASTM) and is closely related to R6, the code
used in the UK nuclear industry to assess the safety of structures with defects. BS 7910
emphasises the need for NDT to detect cracks and also provides guidance on safety
factors, reliability factors and probabilistic methods. The first procedures were written
as a ‘published document’ of the British Standards Institute in 1980 (PD 6493) but
these have now been revised and updated as a British Standard, BS 7910, which was
released in 2001.
Three levels of treatment of flaws are provided. Level 1 is a conservative preliminary
procedure which is very easy to apply; Level 2 is the normal procedure and is more
complicated than Level 1. It contains two sub-options, Level 2A and Level 2B. Level 3
is the most advanced treatment and contains three sub-options, Level 3A, 3B and 3C.
Level 3 is mainly used for ductile materials which exhibit some stable amount of crack
growth before fracture.
All levels use the concept of a failure assessment diagram (FAD) similar to the idea
of a yield surface in plasticity. If an assessment point lies within the diagram, the flaw
or crack is deemed to be safe. If it is outside the diagram it is deemed unsafe and action
must be taken. Note that a crack which fails by the conservative Level 1 option may be
safe when the more accurate Level 2 analysis is carried out. Similarly, a crack which is
unsafe by Level 2 may be safe under Level 3, if a small amount of stable ductile crack
growth is allowed for.
4.1 The failure assessment diagram
Before discussing the procedure in detail a number of issues relevant to each level are
discussed. The philosophy behind the procedure is that failure can occur either due to
excessive plastic deformation (plastic collapse) or by fracture. It is now well known that
these failure modes are not decoupled as fracture and deformation are closely linked,
but the philosophy is maintained for historic reasons and for simplicity of presentation
of the method. The fracture parameters used are K, δ (CTOD) and, for Level 3C, J .
Although K is used in the Level 2 and Level 3A and 3B assessments, they are in fact
elastic-plastic based assessment procedures.
74
The proximity to fracture and plastic collapse are specified by the ratios, Kr (or√δr) and Lr (Sr for Level 1) respectively, where,
Kr =K
KIC
δr =δ
δc
Lr =σref
σy=
P
PL
Sr =σref
σf=
P
PL(σf ).
In the above, σref is the reference stress defined in section 2.5.2. PL is the plastic
collapse load and σf , used in the definition of Sr, is the flow stress—defined in BS 7910
as the lower of 1.2σy or (σy + σu)/2, where σu is the tensile strength of the material.
When defining Sr using the limit load, the flow stress, σf , rather than the yield stress
is used. Hence the notation PL(σf ) above. The use of the flow stress takes account of
the fact that the material strain hardens so that plastic collapse does not occur when
the stress reaches yield.
The use of the square root sign in the quantity√
δr is for consistency with the K
expression, i.e.,
√δ ∝
√J
σy=
K√Eσy
,
and√
δc ∝√
JIC
σy=
KIC√Eσy
.
Hence √δ
δc=
K
KIC,
though this relationship does not hold precisely at high values of σref as discussed later.
In order to determine the linear elastic stress intensity factor, K, the stress in the
uncracked body and the appropriate shape factor, Y must be known. If the body is
of complicated shape then a finite element analysis may be required to determine the
stresses. Using these stresses, K may be obtained from handbook solutions, many of
which are provided in BS7910.
It often proves convenient to linearise the stresses, since K solutions are typically
available only for cracks under tension (constant stress) or bending (linear variation of
75
stress), but not for arbitrary non-linear stress distributions. BS7910 provides guidance
on linearising the stress fields as shown in Fig. 4.1. It is these linear stress distributions
which are used to obtain the K value.
Figure 4.1 Linearisation of stress distributions for (a) surface flaws (b) embedded flaws
The procedures assume that the crack is normal to the maximum principal applied
stress (i.e. it is a Mode I crack), and most of the available K and limit load solutions
are for Mode I loading. However, if, for example, the crack follows a weld boundary (see
below) then it will not be a Mode I crack. In this situation, the crack is projected on the
plane of principal stresses (see Fig. 4.2) which, in combination with other assumptions
in the procedure, is designed to give a conservative assessment.
Figure 4.2 Treatment of an inclined crack within BS7910
However, if the angle between the crack plane and the principal plane is greater than
76
20◦ then mode mixity effects may become important and they must be accounted for.
Advice for this situation is provided in the standard.
Each level of the failure assessment procedure is now discussed in turn.
4.2 Level 1 Failure Assessment Diagram
This is a very simple procedure. The failure assessment diagram (FAD) is shown in
Fig. 4.3. It specifies that the applied load must be less than 80% of the plastic collapse
load, based on the flow stress and K must be less than KIC/√
2. The latter inequality
approximately corresponds to a factor of safety of two on crack length assuming all linear
elastic behaviour—the value of K in Kr is determined from available linear elastic K
solutions.
Figure 4.3 The BS7910 Level 1 Failure Assessment Diagram
If the CTOD approach is to be used, then δ is determined from K by
δ =K2
σyEF (σ/σy),
where F (σ/σy) is an adjustment factor which accounts for plasticity effects on the
CTOD. It is given by the following expression,
F =
1 for σ ≤ 0.5σy
(σ
σy
)−2 (σ
σy− 0.25
)for σ > 0.5σy.
The largest value of F will be when σ/σy = 0.8 (since Sr is always less than 0.8
for a Level 1 assessment) For this case F = 0.86, so in general the correction factor
77
will be small. The above expression for F is for steel and aluminium alloys. For other
materials, F is always taken to be 1.
Note that, in the above, different results would be obtained from a K-based fracture
assessment and a CTOD-based analysis if σ/σy > 0.5. This is a consequence of the fact
that originally these were separate procedures used in different applications and in
bringing the two together in a single standard, some inconsistencies and compromises
are inevitable.
4.3 Level 2 Failure Assessment Diagram
This is the normal assessment procedure for general analysis which is further sub-
divided into Level 2A and Level 2B. It is an elastic-plastic J-based approach so some
background into the derivation of the FAD is provided next. Note that this background
knowledge is not required to apply the method and application of Level 2A is no more
complicated than that of Level 1.
4.3.1 Derivation of Level 2 failure assessment diagram
Take as a starting point, the EPRI power law plasticity solution for J ,
J = ασyεya(P/PL)n+1h(a/w;n),
where PL is the limit load, and the stress-strain relationship of the material is given by
ε/εy = α(σ/σy)n.
Recalling the definition of the reference stress, σref = σy(P/PL), and defining the
reference strain as the uniaxial strain corresponding to the reference stress, i.e. for the
power low hardening material,
εref = αεy(σref/σy)n,
we can rearrange the J equation above to get
J = aεrefσrefh(a/w;n).
The advantage of the above equation is that it can be applied to any material,
provided the uniaxial stress strain behaviour is known, and although it is strictly correct
only for power law hardening materials, it has been shown that it provides a conservative
estimate of J for a wide range of materials.
78
It may be seen that the power-law parameter n still enters the equation for J
through the dependence of h on n. However, it is been found for most engineering
materials of interest and provided that the limit load is used as the normalising load
for J estimation, that h is weakly dependent on n (recall Fig. 2.17) e.g. h(a/W ; n) ≈h(a/W ; 1).
The next stage is to rewrite the above equation in the form of a FAD. The condition
for a safe assessment is simply
J ≤ Jc,
where Jc is the valid plane strain fracture toughness. In other words,
J(σref) ≤ Jc =K2
c
E′ ,
or invertingE′
K2c
≤ 1J
.
Multiplying across by Je = K2/E′ we get failure when
(K
Kc
)2
=Je
J,
where K is simply the linear elastic stress intensity factor. We can rewrite in terms of
Kr,
Kr =K
Kc=
(J(σref)
Je
)−1/2
.
In other words, the curve for the FAD can be determined directly from the rela-
tionship between the total J and Je. This provides us with a definition for the FAD in
terms of σref . Defining Je using the reference stress equation, we get
Je = aσ2
ref
Eh(a/W ; 1),
since for a linear elastic material, α = 1 and εref = σref/E. Then we have
J
Je=
Eεrefσref
,
making the assumption that h(a/W ; n) = h(a/W ; 1), and then
Kr =(
J
Je
)−1/2
=(
Eεrefσref
)−1/2
(∗).
79
Since Lr = σref/σy and εref can be obtained in terms of σref via the uniaxial
stress strain law, this allows a material dependent but geometry independent FAD to
be plotted in terms of Kr and Lr.
The above equation is correct for linear elastic materials (trivial since the result will
be J/Je = 1) and gives a good representation under large scale plasticity. However, in
the intermediate region when there is no remote plasticity, σref < σy, but local yielding
occurs at the crack tip, the agreement with the exact J solution is poor. Therefore an
additional estimate for J is required in this region. For this we used the plastic zone
correction approach.
4.3.2 Estimate of K and J using a plastic zone correction
As discussed in Section 2, when the plastic zone size is relatively small modifications
to the LEFM K are sufficient to account for the effects of material non-linearity. An
effective K is defined,
Keff = Y (ae)σ√
ae
where Y is the LEFM geometry factor. The effective crack length ae is given by,
ae = a + ry,
where a is the actual crack length and ry is half the plane strain or plane stress plastic
zone size as defined in Section 2.5.7.1.
For a small crack under tension it was shown that,
Keff =√
πσ√
a√1− 1
2
(σ
σy
)2.
Replacing σ by σref so the expression is applicable to general geometries and writing in
the form required for an FAD, we get
J
Je=
(Keff
K
)2
=1
1− 12
(σref
σy
)2 .
and using the binomial theorem for small σref/σy we get
J
Je= 1 +
12
(σref
σy
)2
(#).
80
Note that we have used the equation J = K2eff/E′ to determine J from K . This
expression is valid as we are still within small scale yielding conditions.
4.3.3 Overall Level 2 Failure Assessment Diagram
Finally, combining the two expressions for J/Je , Eq. (#) above and Eq. (*) from page
81 into a single expression which reduces to the above when σref < σy and to the earlier
expression for large scale plasticity, the following unified expression is obtained:
J
Je=
Eεrefσref
+12
(σref
σy
)2 (Eεrefσref
)−1
.
The term Eεref/σref appears in both of the terms on the RHS of the equation.
For σref < σy, εref = σref/E so Eεref/σref = 1 and the small scale yielding equation is
obtained. Under large scale plasticity it may be seen that Eεref/σref ≈ εpl/εel which
becomes very large under large scale plasticity and hence the second term becomes
negligible and the equation reduces to the first of the two J expressions.
Writing the above equation in terms of Lr, the FAD becomes,
Kr =(
J
Je
)−1/2
=(
EεrefLrσy
+12
L3rσy
Eεref
)−1/2
.
This, finally, is the form of the FAD for a Level 2B assessment.
To use a CTOD approach Kr is replaced by√
δr as in Level 1 and the CTOD, δ,
is obtained from K again as in Level 1 with some modification to account for plane
stress/plane strain conditions. The details of the CTOD approach are given in the
standard.
The above FAD will depend on the material in question, i.e. a different FAD will
be needed for each material that is being assessed. Figure 4.4 shows a typical Level 2B
FAD for a particular material.
4.3.4 The Level 2A Failure Assessment Diagram
Also shown in Fig 4.4 is the Level 2A FAD which is a lower bound (i.e. conservative)
Level 2 FAD for a wide range of materials. If uniaxial stress-strain data is unavailable
this FAD may be used. The equation for the Level 2A FAD is
√δr or Kr = (1− 0.14L2
r )(0.3 + 0.7 exp (−0.65L6
r ))
The Level 2A FAD is a geometry and material independent curve which makes it
very simple to use.
81
Note that neither Level 2A nor Level 2B are exact solutions but Level 2B is closer
to reality than Level 2A. For a wide range of cases, it has been shown by comparing
with numerical solutions and experiments that both curves are conservative and Level
2A is more conservative than Level 2B.
4.3.5 Definition of cutoff line
In Fig. 4.4 it may also be seen that there is a cutoff at a point defined as Lrmax.
This cut-off which depends on the material is to prevent global plastic collapse. The
cutoff is defined as
Lrmax =σf
σy,
where σf is the flow stress. If Lr > Lrmax then the crack is unsafe, regardless of the
value of Kr.
Figure 4.4 BS7910 Level 2 Failure Assessment Diagrams (a) Level 2A FAD, (b) Level
2B FAD derived from material stress/strain data of (c).
4.3.6 Use of the Level 2 procedure
The use of the procedure is exactly as for Level 1, i.e. Kr and Lr are determined
from handbook solutions or numerical calculations (elastic calculations for Kr, perfectly
82
plastic for Lr) and the point located on the FAD. If the point lies inside the FAD then
the crack is safe.
4.4 Level 3 procedure
The Level 3 procedure is divided into three methods. Level 3A and 3B are essen-
tially the same as Level 2A and 2B respectively, apart from the fact that the resistance
curve is used to determine the material parameter Kc. Therefore, even if the initial
point lies outside the FAD, provided there is a sufficient increase in the J-resistance
curve, after a small amount of ductile tearing, the crack may be safe.
Level 3C involves the use of a further, more accurate FAD, determined from a full
elastic-plastic J analysis of the structure. The FAD is then determined directly from
Kr =(
J
Je
)−1/2
and the assessment is carried out as for Level 3A and Level 3B. There is no reason why
the Level 3C FAD could not be used for a Level 2 type analysis, i.e. obtaining the FAD
directly from J and using the initiation KIC value, rather than the enhanced toughness
after some amount of crack growth. However, it is assumed that if one is going to the
expense of a full numerical analysis, one will also want to take full advantage of the
material toughness.
83
5. Creep Fracture Mechanics
Creep occurs when a component is held under stress over long times or high tempera-
tures or a combination of both (see 2M Materials’ notes). Creep is a time dependent
process that results in permanent (non-recoverable) deformation and may ultimately
lead to failure (creep rupture). Creep is particularly important in chemical process
plant, electrical power generation equipment and aircraft gas turbine engines and often
it is failure due to creep that is the predominant design failure mode.
In metals, creep is generally divided into primary, secondary and tertiary creep.
Primary creep is the creep which occurs over short times and results in a decreasing
strain rate. Secondary creep generally occurs over the largest period of a components
lifetime and is characterised by a constant (steady state) creep rate. Tertiary creep
occurs at long times, close to the time of failure and gives a very high creep rate. These
modes are illustrated by the typical creep curve shown in Fig. 5.1.
Figure 5.1, Typical strain vs time creep curve
5.1 Secondary creep
Since a component will spend most of its lifetime in the secondary creep regime, creep
fracture mechanics has focused on this regime. During secondary creep, cracks which
were initially safe, may grow slowly, under a constant load, and lead to failure, a process
analogous to crack growth by fatigue under cyclic loading.
For many materials, the secondary creep deformation behaviour is well charac-
terised by a power law creep relationship, analogous to power law plasticity,
εc
ε0=
(σ
σ0
)n
,
84
where n, σ0 and ε0 are material constants. This equation is identical to the power law
plasticity relationship but with ε0 replacing ε0 and α = 1. It may therefore be shown
that solutions to power law plasticity are also solutions to power law creep except that
displacement rate and strain rate replace displacement and strain, respectively, in the
creep solution. Therefore, we immediately have the solution to the steady state creep
stress and strain rate at the crack tip for a power law creep material, i.e.
σij/σ0 =(
C∗
ε0σ0Inr
)1/(n+1)
σij(θ; n)
εcij/ε0 =
(C∗
ε0σ0Inr
)n/(n+1)
εij(θ; n).
The parameter C∗ is the creep analogy to J and is defined in the same way as J using
a contour integral,
C∗ =∫
Γ
W (ε)dy − t∂u∂x
ds,
where
W (ε) =∫ ε
0
σdε
is the strain energy rate density. This is a path independent integral (the proof follows
immediately from the path independence of the J integral). However, the path inde-
pendence relies on the fact that steady state conditions are prevailing, i.e. the creep
strain rates are much larger than the elastic strain rate in the body. This will hold after
long times and when the remote applied load is constant.
5.2 Estimation of C∗ in specimens and components
The same procedures developed to estimate J can be used to estimate C∗ under
steady state conditions. For example C∗ can be estimated from load-displacement rate
data,
C∗ =η
B(W − a)A,
where A is the area under the load displacement-rate curve. Note that analogous to
pure power law plasticity,
A =n
n + 1P ∆,
where ∆ is the remote displacement rate corresponding to the load P , allowing the
above equation to be simplified to
85
C∗ =η
B(W − a)n
n + 1P ∆.
Values of η for various crack geometries are available in the fracture mechanics hand-
books or the ASTM standard—they are identical to the elastic-plastic η factors.
In the same way, the GE-EPRI scheme developed to estimate J can be used to
estimate C∗,
C∗ = aε0σ0
(P
P0
)n+1
h(a/w; n).
Note that in creep, P0 no longer has the interpretation of a limit load. It is simply
a normalising load, based on the material parameter σ0. More commonly in the UK,
the reference stress approach, which is an extension of the GE-EPRI approach is used to
estimate C∗ because of its simplicity. Recall from Section 4.3.1 that under pure power
law plasticity,J may be estimated using the following equation,
J
Je=
Eεrefσref
,
with σref defined via the plastic collapse load.
Similarly for power law creep, we can write,
C∗
Je=
Eεrefσref
.
It is convenient in the creep case to replace Je by K and the equation becomes
C∗ =K2εrefσref
,
and we then write
C∗ = σref εrefR′,
where R′ is a length scale which depends only on the geometry of the component,
R′ = (K/σref)2, e.g. for a crack in an infinite plate R′ = πa. Once again it needs to be
emphasised that this equation gives an estimate of C∗ and to obtain an exact value a
study of the actual cracked geometry in question is needed.
5.3 Creep solutions for short times
The above solutions are for long times when the stress and strain rate fields are
constant throughout the structure. The period between initial loading and final steady
86
state is called the redistribution period. Figure 5.2 shows how stresses redistribute
during creep from the initial elastic K field at t = 0 to the steady state value at long
times.
/
Figure 5.2 Redistribution of stresses during creep
In the analysis shown in Fig. 5.2 the power law creep exponent, n was equal to 3.
Therefore the steady state creep distribution is quite close to the elastic distribution
(n = 1). The higher the value of n the lower the crack tip fields so for n = 10 more
redistribution of stress will take place.
During the redistribution period, the global elastic and creep strain rates are com-
parable. However, because of the power law nature of creep and the large stress gradients
near the crack tip, there will still be a region in the vicinity of the crack where the creep
strain rates are much higher than the elastic strain rates and the HRR-type solution
discussed above applies. We write,
σij/σ0 =(
C(t)ε0σ0Inr
)1/(n+1)
σij(θ; n)
εcij/ε0 =
(C(t)
ε0σ0Inr
)n/(n+1)
εij(θ; n),
where the notation C(t) is used to emphasise that the amplitude of the stress depends on
time. The zone of dominance of the HRR-type field may be very small—it approaches
zero at very short times (when the K field dominates).
The equation for C(t) is identical to that for C∗ the only difference being that C(t)
is no longer a path independent integral and can only be defined asymptotically, i.e.
the contour Γ very close to the crack tip,
87
C(t) =∫
Γ→0
W (ε)dy − t∂u∂x
ds.
Figure 5.3 shows the variation of C(t) with time determined from a numerical
analysis. The decreasing magnitude indicates that the amplitude of the crack tip stress
field reduces with time due to creep redistribution (see also Fig. 5.2),
Figure 5.3 Variation of C(t) with time, (Webster and Ainsworth).
After some time the value of C(t) approaches the steady state C∗ value. The normalising
time, tT, in Fig. 5.3 is an approximation to the time required to reach steady state and
is called the transition time (there is also a quantity called the redistribution time which
provides a slightly improved estimate of time to reach steady state). The equation for
tT is
tT =K2
(n + 1)E′C∗.
For the case illustrated in Fig. 5.3 it may be seen from the numerical result, that
the time to reach steady state tss ≈ 3tT. The approximate solution to C(t) shown in
the figure is given by the equation,
C(t) =(
tTt
)C∗ =
K2
(n + 1)E′t,
and it may be seen that
C(tT) = C∗.
Note also that the above equation gives the physically unrealistic solution that C(t) = ∞at t = 0. There are other more complicated equations to estimate C(t), but creep
fracture mechanics is still primarily based on C∗.
88
Note that the stress and strain rate fields given above are strictly speaking appli-
cable only for stationary cracks and under creep conditions the crack will be growing.
However, because the crack is growing very slowly, there is sufficient time for the station-
ary crack distributions to re-establish themselves after each increment of crack growth.
It should also be noted that even under constant load the value of C∗ will be increasing
as the crack length will be increasing. So for example, in the reference stress estimate
for C∗ both σref and R′ will increase (slowly) with time.
5.4 Characterisation of creep crack initiation and growth
Under a constant load and at high temperature, a pre-existing crack will grow
slowly due to creep until final failure. Generally, failure due to creep is by a stable
ductile process, involving growth and coalescence of micro-voids. Since under steady
state conditions C∗ characterises the stress and strain rate at the crack tip, it is to be
expected that C∗ will also provide a good measure of the crack growth rate under creep
conditions. A typical curve is shown in Fig. 5.4 where the rate of crack growth, a, is
plotted against C∗ for two specimen geometries of an alloy steel. It may be seen, that
within the scatter of the data, the creep crack growth rate for the steel is characterised
by C∗.
Figure 5.4 Crack growth rates, a, plotted against C∗ for two specimen types
Note that a single test can generate a full set of a vs. C∗ data as the crack is
growing during the test. Typically, in such a test, C∗ is estimated using the load-line
displacement rate and the rate of increase of crack length a is estimated using visual
inspection, compliance methods or ‘potential drop’ techniques, whereby a constant cur-
89
rent is applied across the crack plane and the potential drop is correlated with the
increase in crack length. In the latter two methods, comparison with the final crack
length determined from heat tinting and breaking open of the specimen should be used
to check the predicted amounts of crack growth.
5.4.1 Model for steady state creep crack growth
We consider a crack which is assumed to be growing at a rate, a with a steady
state creep field characterised by C∗. It is assumed that the process of crack growth is
a strain controlled process with material failure occurring at a point initially a distance
rc ahead of the crack tip, when a strain εf is reached (see Fig. 5.5).
Figure 5.5 Distribution of creep strain rate ahead of a crack in a creeping material
Given the expression for the creep strain rate at a distance r from the crack tip,
εcij = ε0
(C∗
ε0σ0Inr
)n/n+1)
εij(θ;n),
the creep strain accumulated over a time t is given by
εc =∫ t
0
εcdt = ε0ε(0; n)(
C∗
ε0σ0In
)n/n+1 ∫ t
0
1r(t)n/n+1
dt,
where εc is the equivalent (von Mises) creep strain, and the dependence of distance, r,
on time, due to the movement of the crack, is emphasised. It is assumed in the above
equation that creep crack growth occurs directly ahead of the crack tip (θ = 0) but it is
not difficult to generalise the equation to allow creep crack growth at any angle. Using
a change of variables in the above,
εc = ε0ε(0; n)(
C∗
ε0σ0In
)n/(n+1) ∫ 0
rc
1rn/n+1
dt
drdr.
Sincedt
dr= 1/
dr
dt= −1
a,
90
we can write
εc =ε0ε(0; n)
a
(C∗
ε0σ0In
)n/(n+1) ∫ rc
0
1rn/n+1
dr
= (n + 1)ε0ε(0; n)
a
(C∗
ε0σ0In
)n/(n+1)
r1/(n+1)c .
Crack growth occurs when this strain is equal to the material ductility, εf . Note that as
discussed previously, the ductility depends on the triaxiality, (σm/σe). We may rewrite
the above equation as an equation for crack growth rate, a,
a = (n + 1)ε0ε(0; n)
εf
(C∗
ε0σ0In
)n/(n+1)
r1/(n+1)c .
The distance rc is sometimes identified as the creep process zone size, i.e. the
distance ahead of the current crack tip where creep damage is significant. Note that
because rc is raised to the power 1/(n + 1), the dependence of a on the value of rc
is weak, so its value is not very significant. However, the crack growth rate depends
inversely on the ductility, εf . The triaxiality, σm/σe, under plane strain conditions is
about a factor of three higher than plane stress, leading to a creep failure strain εf
about 30 times lower than in plane stress, using a void growth criterion, as discussed
earlier. Thus crack growth rates are expected to be about 30 times higher under full
plane strain conditions than under plane stress conditions at the same value of C∗.
The important point is that there is a one-to-one relationship between a and C∗
and that crack growth rate is given by an equation of the form,
a = A(C∗)φ.
The above form has been confirmed by experiment and leads to a straight line on
a log-log plot. A general equation has been proposed by Nikbin, Smith and Webster to
cover a wide range of n values. They have shown that the theoretical equation derived
earlier can be simplified by follows,
a =3(C∗)0.85
εf.
This equation has been shown to predict crack growth rates to within a factor of
two for a wide range of materials. In the above equation, εf is the appropriate ductility,
i.e. equal to the uniaxial creep ductility under plane stress (i.e. thin components) and
equal to 1/30 times the uniaxial ductility under plane strain (thick components). A
91
comparison of these two crack growth rate equations with the earlier experimental data
is shown in Fig. 5.6. The increased creep rates predicted by the plane strain solution
may be seen. It appears that the data for the tests in Fig. 5.6 correspond more closely
to plane stress conditions.
Figure 5.6 Comparison of theoretical creep crack growth equations with experimental
data
5.4.2 Creep Initiation
The equations derived earlier have given the rate of crack growth under secondary
creep. However, recently there has been increased interest in predicting the incubation
period, i.e. the amount of time before a crack starts to extend under creep conditions.
This can be determined in a very similar manner to the creep crack growth rate.
We assume again, that the crack tip fields are given by C∗ and the HRR field. Crack
initiation occurs when the strain at some critical distance from the crack tip reaches the
material ductility. This critical distance, may be a material distance like a grain size or
simply the resolution of the crack growth detecting device. As before
εc =∫ t
0
εcdt = tεc.
Note that the above assumes that even during incubation, most of the strain accumu-
lated is during the steady state period, i.e. C(t) = C∗ and therefore the strain rate is
constant during incubation (there is no theoretical difficulty in allowing C(t) to vary
92
with time and include this in the integration.) The analysis for initiation differs from
the crack growth analysis in that in this case the crack tip is assumed stationary.
Using the above we get,
εc = tε0ε(0; n)(
C∗
ε0σ0Inrc
)n/n+1
and
ti =εf
ε0ε
(ε0σ0Inrc
C∗
)n/n+1
and again the dependence of incubation time on C∗ is illustrated.
The quantities in the above equation are not always easily obtainable. A good
estimate of the incubation time has been shown to be given by the following, reference
stress based, expression:
ti = 0.0025[σref trK2
]0.85
,
where tr is the time to rupture in a uniaxial test carried out at σ = σref . In the above
equation, units of stress are in MPa, time in hours and K in MPa√
m.
5.5 Elastic-plastic creep
In the preceding analyses, it has been assumed that the plastic strains are negligible.
(The distinction made here between rate independent ‘plasticity’ and rate dependent
‘creep’ is somewhat questionable, but it suffices to explain most high temperature frac-
ture phenomena.) Incorporating the effect of plasticity is not difficult: the initial stress
field is then given by the elastic-plastic HRR field, rather than the K-field and, as for
the elastic-creep case, the stress fields will redistribute until steady state conditions are
reached. Usually the redistribution times are shorter, as the elastic-plastic stress distri-
bution are closer to the creep distributions than the elastic ones. The equations derived
previously still hold with the requirement that for steady state,
εc > εe + εp,
where εe and εp are elastic and plastic strain rates, respectively.
In the derivation of creep crack growth laws, a vs C∗, and incubation times it is
generally assumed that damage due to plastic strain is of a different type than that due
to creep strain and hence the contribution from the plastic strain is not included in the
total strain required to cause crack initiation or growth.
93
5.6 Micrographs of creep failure
As discussed previously, creep failure occurs primarily in a ductile manner. Under
creep conditions voids typically nucleate on grain boundaries or triple grain junctions
as shown in Fig. 5.7(a). Final failure is then associated with linking up of microcracks
or voids along grain boundaries—intergranular fracture as shown in Fig. 5.7(b).
Figure 5.7 Damage observed in materials loaded under creep conditions
94
6. Appendices
6.1 Appendix A, Extracts from two key papers on non-linear fracture me-
chanics
“Fundamentals of the phenomenological theory of nonlinear fracture mechanics” by
J.W. Hutchinson, Journal of Applied Mechanics, Vol. 50, 1983.
“On macroscopic and microscopic analyses for crack initiation and crack growth tough-
ness in ductile alloys” by R.O. Ritchie and A.W. Thompson, Metallurgical Transactions
A, Vol. 16A, 1985.
95
These pages are not made available on the college web page
96
6.2 Appendix B, List of important equations for Advanced Fracture Me-
chanics
(Appendix B and C may be used freely in exams)
Page 1 of 4
Strain energy density, W, for a linear elastic material under uniaxial stress, σ :
E2W
2σ=
Strain energy density, W, for a power law material under uniaxial stress, σ :
σε1n
nW
+=
Definition of strain energy Ue:
∫=V
e WdVU
Strain energy for a linear elastic material under an applied force, P, giving rise to a displacement, ∆ :
2
PUe
∆=
Strain energy for a linear elastic material under an applied moment, M, giving rise to a curvature, θ :
2
MUe
θ=
Strain energy for a power law material under an applied force, P, giving rise to a displacement, ∆:
∆+
= P1n
nUe
External work, Wext, for a moment, M, giving rise to a curvature, θ:
Wext = Mθ.
External work, Wext, for a force, P, giving rise to a displacement, ∆:
Wext = P∆.
Definition of energy release rate G,
a
U
B
1G
δδ−=
Energy release rate-compliance relation for a linear elastic material:
da
dCP
B2
1G 2=
Definition of K for a cracked geometry:
aYK σ= ;
Center crack in an infinite plate under tension, Y = √π ;
Edge crack in an infinite plate under tension, Y = 1.12√π.
97
Page 2 of 4
rp
Crack opening displacement for a Mode I crack in a linear elastic material:
( )π2
r
E
K8ru
′=∆ ; strain.planeforstress;planefor
21
EEEE
υ−=′=′
Relationship between energy release rate and stress intensity factor:
modulusshear ;2
222
=+′
+= GG
K
E
KKG IIIIII
Phase angle, I
II
K
K1-tan=φ ; Mode mixity, φπ2
M = .
Branching angle, θ, for a crack under mixed mode conditions in a linear elastic material, using the maximum hoop stress criterion:
( ) �����
�����
+�������
�−= 8
K
K
K
K
4
12
2
II
I
II
Iθtan ,
Plastic zone size, rp, ahead of a Mode I crack:
strain planefor 3
1 stress planefor
1;
2
πα
πα
σα ==���
��
= ;K
ry
p
Plastic zone correction for K:
( ) 2/raa;aaYK peee +== σ
ASTM size requirement for KIC testing:
a, b, B > 2.5(KIC/σy)2.
Definition of J line integral:
dsx
WdyJ �Γ
−=δδu
t
HRR crack tip stress and strain distribution for a power law hardening material:
( ) ( )nrI
Jn
rI
Jij
nn
n
ijij
n
n
ij ;~;;~)1/(
000
)1/(1
000
θεσαεαε
εθσ
σαεσσ ++ ������=
������=
Relationship between crack opening displacement, δ, and J integral:
0n
Jd
σδ = ; strain. planefor 5.0 stress; planefor 1 ≈≈ nn dd
θ
x
y
Γ
98
Page 3 of 4
Plane stress limit moment for a shallow cracked beam in bending:
( )4
aWBM
2
yL−= σ
Plane stress limit load for a shallow cracked beam in tension:
( )aWBP yL −= σ
Evaluation of J using the η factor:
( ) p
2A
aWBE
KJ
−+
′= η
; η ≈ 1 for tension loading; η ≈ 2 for bend loading.
Evaluation of J using GE-EPRI equation:
( )n;W/ahP
Pa
E
KJ
1n
000
2 +
����
����+′
= εασ
ASTM size requirement for JIC testing (for deep cracked bend specimen)
a, b, B > 25(JIC/σy)
Rice and Tracey void growth relation:
( ) ∞= pym /..r
r εσσ ��51sinh5580
BS7910 Failure Assessment Diagram:
( )fLf
refr
Ly
refr
ICr P
PS
P
PL
K
KK
σσσ
σσ
===== ;;
( ) 2/2.1;P
Puyyfy
Lref σσσσσσ +=���
��
= or
Failure condition:
2/1
er
J
JK
−����
�=
BS7910 Level 2A FAD, ( ) ( )( )6r
2rr L65.0exp7.03.0L14.01K −+−=
BS7910 Level 2B FAD,
2/1
ref
y3r
yr
refr
E
L
2
1
L
EK
− ��
����
+=ε
σ
σ
ε
99
Page 4 of 4
y
fmaxrL
σ
σ=
Definition of C* line integral:
dsx
dyW*C ∫Γ
−=δδu
t�
�
Evaluation of C* for a power law material using the η factor:
( )∆
+−= �P
1n
n
aWB*C
η
Evaluation of C* using GE-EPRI equation:
( )n;W/ahP
Pa*C
1n
000
+
= σε�
Equation for C* based on σref:
R*C refref ′= εσ � ;
2
=′
ref
KR
σ
Creep crack growth rate equation:
( )φ*CAa =�
100
6.3 Appendix C, Linear Elastic K field distributions
(Appendix B and C may be used freely in exams)
101
102
103
