Advanced Fracture Mechanics: Lectures on Fundamentals of Elastic, Elastic-Plastic and Creep Fracture, 2002–2003

107 Pages·2003·3.679 MB·English

by  Noel O’Dowd

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
CONTENTS
Page
Course Description .............................................................. ii
Introduction .................................................................... iii
1. Linear Elastic Fracture Mechanics........................................ 1
1.1 Definition of energy release rate, G ......................................... 1
1.2 Strain energy, energy release rate and compliance........................... 3
1.3 Stress analysis of cracks.................................................... 8
1.4 Mixed mode fracture mechanics........................................... 20
1.5 Concept of small scale yielding............................................ 29
2. Non-linear Fracture Mechanics ......................................... 30
2.1 The J integral, (Rice, 1968) .............................................. 30
2.2 Power law hardening materials—The HRR field ........................... 39
2.3 Crack tip opening displacement ........................................... 42
2.4 Relationship between J and G............................................. 43
2.5 Evaluating J for test specimens and components .......................... 45
2.6 Application of non-linear fracture mechanics............................... 57
2.7 K dominance, J dominance and size requirements......................... 60
2.8 Standard test to determine J ........................................... 64
IC
3. Micromechanisms for ductile and brittle fracture...................... 66
3.1 Micromechanism of cleavage failure ....................................... 66
3.2 Prediction of fracture toughness using the RKR model and the HRR field.. 67
3.3 Micromechanism of ductile failure......................................... 68
3.4 Prediction of fracture toughness using the MVC model and the HRR field . 69
3.5 Competition between brittle and ductile fracture .......................... 72
4. Application of BS 7910 in failure assessments ......................... 74
4.1 The failure assessment diagram ........................................... 74
4.2 Level 1 Failure Assessment Diagram....................................... 77
4.3 Level 2 Failure Assessment Diagram....................................... 78
4.4 Level 3 Procedure......................................................... 83
5. Creep Fracture Mechanics ............................................... 84
5.1 Secondary creep........................................................... 84
5.2 Estimation of C∗ in specimens and components............................ 85
5.3 Creep solutions for short times............................................ 86
5.4 Characterisation of creep crack initiation and growth...................... 89
5.5 Elastic-plastic creep....................................................... 93
5.6 Micrographs of creep failure............................................... 94
6. Appendices ............................................................... 95
6.1 Appendix A, Extracts from two key papers on non-linear fracture mechanics
6.2 Appendix B, List of important equations for Advanced Fracture Mechanics
6.3 Appendix C, Linear Elastic K field distributions
i
September 2002
Imperial College London
Department of Mechanical Engineering
4M/AME Advanced Fracture Mechanics
The course, which will be given by Dr O’Dowd, consists of approximately 22 lectures and
9 tutorials. Examination will be by written paper at the end of the course and by problem
sheets (5 in total) which will be distributed throughout the year and are worth 20% of the
total course mark. 4M students must have taken the course 3M course, “Fundamentals of
Fracture 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 describing
failure in engineering components. In addition, the course will explain how to apply these
concepts 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 exercises
provided 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 considered
background reading and are not required texts. The texts are listed in alphabetical order.
1. T.L.Anderson‘Fracturemechanics: fundamentalsandapplications’, EdwardsArnold,
London, 1991.
2. R.W.Hertzberg, ‘Deformationandfracturemechanicsofengineeringmaterials’, Wiley
and Sons, New York, 1989.
3. M.F. Kanninen and C.H. Popelar, ‘Advanced fracture mechanics’, Oxford University
Press, 1985.
4. G.A. Webster and R.A. Ainsworth, ‘High temperature component life assessment’,
Chapman and Hall, London, 1994.
ii
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.
iii
1. Linear elastic fracture mechanics
1.1 Definition of energy release rate, G
Griffith (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.
B DS =g (dA)
s
D a
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 = δAγ where δA is the new surface area created
s
and γ is the surface energy per unit area, as illustrated above. The change in area
s
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
1 ∆U
− = 2γ .
s
B ∆a
Rewriting as a partial derivitive we get Griffith’s relation,
1 ∂U
− = 2γ
s
B ∂a
1
If this equation is satisfied then crack growth will occur. The energy release rate, G is
defined as
1 ∂U
G = − .
B ∂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, γ , depends only on
s
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 ≥ 2γ = G .
(cid:124)(cid:123)(cid:122)(cid:125) s (cid:124)(cid:123)(cid:122)c(cid:125)
crack driving force 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 2γ for most
s
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 = γ +γ ,
s p
whereγ istheplasticworkdissipatedinthematerialperunitcracksurfaceareacreated
p
(in general γ >> γ ). Then the criterion for fracture becomes
p p
1 ∂U
− ≥ 2(γ +γ )
s p
B ∂a
or
1 ∂U
G = − ≥ 2(γ +γ ) = G .
s p c
B ∂a
The Griffith and Irwin/Orowan approaches are mathematically equivalent, the only
difference is in the interpretation of the material toughness G . In general G is obtained
c c
2
directly from fracture tests which will be discussed later and not from values of γ and
s
γ . The critical energy release rate, G , can be considered to be a material property
p c
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,
(cid:90) ε
W = σdε,
0
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.
s, e
(cid:0)
(cid:1)(cid:3)(cid:2)
s
(cid:1)
W
e
(cid:0)
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.
3
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.
s loading
(cid:0)
s loading
(cid:0)
unloading
Work done
unloading
Elastic Material
Elastic-plastic
Work done = 0
material
e
(cid:1)
e
(cid:1)
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.
1
W = σε.
2
In uniaxial loading
σ2
W = .
2E
4
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,
n
W = σε.
n+1
1.2.3 Definition of Strain energy, U
e
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,
(cid:90)
U = WdV,
e
V
where V is the volume of the body. The strain energy, U , is the strain energy density
e
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,
P∆
U = ,
e
2
where P and ∆ are the applied load and conjugate displacement. For a power law
hardening material it can be shown that,
n
U = P∆.
e
n+1
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.
D
Figure 1.4, Schematic of a loaded cracked body.
5
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 = U
e
For prescribed load, P: U = U −P∆
e
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:
P∆ P∆
U = −P∆ = −
2 2
(cid:181) (cid:182) (cid:181) (cid:182)
1 ∂U P ∂∆
G = − =
B ∂a 2B ∂a
P P
(cid:181) (cid:182)
∂
where the notation emphasises that load P is held constant.
∂a
P
∆ = C(a)P
(cid:181) (cid:182)
∂∆ dC
⇒ = P
∂a da
P
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