Advanced fracture mechanics

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 JIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 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

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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 understanding 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, ductile and creep conditions. Lectures are presented on the underlying principles and exercises provided to give experience of solving practical problems. Objectives 1. 2. 4. 5. 6.

At the end of the course the student should be able to: Understand the mechanisms of fracture under brittle, ductile and creep conditions. Explain the relationship between linear elastic and non-linear fracture concepts and the terms K, G, J and C ∗ . Establish the theoretical stress distributions ahead of a crack under brittle, ductile and creep conditions. Appreciate how to make valid fracture toughness measurements on materials. 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 covered 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 ‘Fracture mechanics: fundamentals and applications’, Edwards Arnold, London, 1991. 2. R.W. Hertzberg, ‘Deformation and fracture mechanics of engineering materials’, 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.

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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 relationships 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.

∆S = γ s (δA)

B

∆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 = 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 −

1 ∆U = 2γs . B ∆a

Rewriting as a partial derivitive we get Griffith’s relation, −

1 ∂U = 2γs B ∂a 1

If this equation is satisfied then crack growth will occur. The energy release rate, G is defined as

1 ∂U . B ∂a In almost all situations ∂U/∂a is negative, i.e. when the crack grows the potential G=−

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 ≥ |{z} crack driving force

2γs

=

Gc . |{z} 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 − or G=−

1 ∂U ≥ 2(γs + γp ) B ∂a

1 ∂U ≥ 2(γs + γp ) = Gc . B ∂a

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 2

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, Z ε W =

σdε, 0

where σ is the stress tensor (matrix), ε is the strain tensor (matrix). Under uniaxial loading W is simply the area under the stress-strain curve (note: not the loaddisplacement curve)as illustrated in Fig. 1.2. In general, the strain energy will not be constant throughout the body but will depend on position.

σ,
 ε σ


W

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. 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. σ

loading

σ

loading

unloading

Work done

unloading Elastic Material Work done = 0

Elastic-plastic material





ε

ε

Figure 1.3, Comparison between behaviour of an elastic material (left) and an elasticplastic 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 =

1 σε. 2

W =

σ2 . 2E

In uniaxial loading

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, W =

n σε. n+1

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, Z Ue = W dV , V

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 P ∆. 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. ∆

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 = 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 × displacement) 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)P n , 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= 1 G=− B µ where the notation

∂ ∂a

P∆ P∆ − P∆ = − 2 2 µ

∂U ∂a

¶

¶

P

P = 2B

µ

∂∆ ∂a

¶ P

emphasises that load P is held constant. P

∆ = C(a)P µ ⇒

∂∆ ∂a

¶ =P P

6

dC da

Therefore

1 2 dC P . 2B da

G= If displacement ∆ is held fixed:

U = Ue =

P∆ 2

and it can be shown (this is left as an exercise) that again 1 2 dC P . 2B da

G=

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:

µ

∂G ∂a

¶ = P

1 2 d2 C P 2B da2

For fixed displacement we get µ

∂G ∂a

¶ ∆

1 2 = P 2B

Ã

d2 C 2 − 2 da C

µ

dC da

¶2 ! .

For most geometries, d2 C/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 > d2 C/da2 then ∂G <0 ∂a so G decreases with crack growth and crack growth is stable, i.e. the applied displacement must be increased to maintain crack growth. Criteria for unstable crack growth: G = Gc

and

∂G > 0. ∂a

Stability arguments are important because while an amount of stable crack growth may be acceptable, unstable fracture must always be avoided. 7

Most materials are ductile and even under nominally linear elastic conditions, fracture 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).

R curve Gr No R curve behaviour Gc

∆a 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 dGr > . ∂a 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,



σxx  σ(x, y, z) = σyx σzx 8

σxy σyy σzy

 σxz σyz  , σzz

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).

y

z x

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 ∂σyy ∂σzz = = . . . = 0. ∂z ∂z ∂z 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.

σ θθ

σ

θ

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 condi9

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. ∞ X σ(r, θ) = Ai rλi fi (θ) i=1

where Ai and λi are unknown real constants and fi are unknown functions of θ. If sufficient 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. A σ ∼ √ f (θ) + . . . r 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: KI θ σθθ = √ cos3 + . . . 2 2πr µ ¶ KI θ θ 2 θ σrr = √ cos + cos sin + ... 2 2 2 2πr θ KI θ sin cos2 + . . . τrθ = √ 2 2 2πr 10

Note that at θ = 0,

KI σθθ = σrr = √ ; 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 σrr τrθ

µ ¶ KII θ 2 θ =√ −3 sin cos + ... 2 2 2πr ¶ µ KII θ 3 θ + ... =√ sin − 3 sin 2 2 2πr µ ¶ KII θ θ 2 θ =√ cos − 3 cos sin + ... 2 2 2 2πr

and at θ = 0 σθθ = σrr = 0;

KII τrθ = √ ; 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, ½ σzz =

ν(σrr + σθθ ) plane strain 0 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 θ KIII cos + . . . τzθ = √ 2 2πr KIII θ τzr = √ sin + . . . 2 2πr 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 11

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.

Mode I

Mode II

Mode III Y

Y X

X

Z

Y X

Z

Mode I: Mode II: Mode III:

Opening or Tensile Mode Sliding or In-Plane Shear Mode 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

Z

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, KI ∆u(r) = 2 E where

r

r (1 + ν)(κ + 1), 2π

   3 − 4ν κ= 3−ν   1+ν

plane strain plane stress.

Substituting for κ the above can be rewritten as r KI r ∆u(r) = 8 0 E 2π where 0

E =

  

E 1 − ν2

plane 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 1 σ, ² ∝ √ ; 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, σyy , and shear, τxy and τyz as shown in Fig. 1.10.

σyy

τ yz τ xy

Figure 1.10 An infinite plate with a centre crack of length 2a. It can be shown that,

√ ∞ πσyy a √ ∞√ = πτxy a √ ∞√ = πτyz a

KI = KII KIII

√

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 ∞ ∞ to KII does not contribute to KI or σyy also that the modes are uncoupled, i.e. τxy

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 √ ∞√ a KI = 1.12 πσyy 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 πσb∞ a, where σb∞ is the bending stress, σb∞ =

My W 12 6M =M = 2 , 3 I 2 W b W b

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 )σy∞ a √ ∞ KII = YII (a/W )τxy a √ ∞ KIII = YIII (a/W )τyz 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 centerline, y = 0, is given by ∞ σyy |x| σyy = √ 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

∞ σyy |r + a|

∞ σyy |r + a|

∞ σyy |r + a| =p =√ =√ . r2 + 2ar + a2 − a2 r2 + 2ar (r + a)2 − a2

Focusing on the crack tip, by letting r → 0; then r + a → a and r2 + 2ar → 2ar and we get σyy

√ ∞ ∞ σyy a σyy a ≈√ = √ . 2ar 2r

If we compare this result with the K field for a Mode I crack derived earlier, KI σyy = √ . 2πr Equating the two expressions for σyy we get, KI =

√

√ ∞ πσyy 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 = lim

√

r→0

and similarily KII = lim

r→0

2πr [σyy (θ = 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

State A

σyy

Y X ∆a

State B

σyy

Energy released

X

∆u ∆a

u

∆u

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.) Z

∆a

Energy released (per unit thickness) = 0

1 σyy (x)∆u(x)dx 2

where ∆u(x) is the separation of the crack faces, the crack opening displacement. For Mode I: K σyy (x) = √ 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 0 , 8K(a + ∆a) ∆u(x0 ) = E0

r

−x0 , 2π

(−x0 , because in the equation for crack opening displacement, the distance to the crack tip, r, is always positive.) Since x0 = x − ∆a we can write, 8K(a + ∆a) ∆u(x) = E0 18

r

∆a − x , 2π

and substituting in the above equation we get 2K(a) K(a + ∆a) Energy released = πE 0

Z

∆a

0

r

∆a − x dx. x

Now by definition, energy released (per unit thickness)= G∆a, therefore, r Z 2K(a) K(a + ∆a) ∆a ∆a − x G∆a = dx. πE 0 x 0 Evaluating the integral,

Z

∆a

r

0

π ∆a − x dx = ∆a, x 2

and the equation becomes, G∆a =

K(a) K(a + ∆a) ∆a. E0

Dividing by ∆a and letting ∆a → 0 so K(a + ∆a) → K(a) = K we get G= where (as before)

 

0

E =



K2 , E0

E 1 − ν2

plane 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=

2 2 KI2 + KII KIII + E0 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 = 500M P a and rc = 10µm, then under Mode I loading KI σyy = √ . 2πr At failure KIC σyy = σc | at r = rc = √ 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 generally 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

SFD

BMD

A

B

A

At crack tip: shear force only, no bending => pure Mode II, M = 1

B

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 KI σyy = √ ; 2πr

KII τxy = √ . 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 φ = 0 ⇒ Mode I

KII KI

φ = π/2 ⇒ Mode II

or sometimes as M where

2 φ π

M= M = 0 ⇒ Mode I

(since tan(π/2)= ∞)

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.

σyy σxy θ 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) ¤ KII £ KI cos3 (θ/2) + √ σθ = √ −3 sin (θ/2) cos2 (θ/2) . 2πr 2πr 22

Note that the hoop stress is normal to any branched crack.

σθ

θ Figure 1.18 Hoop stress, σθθ at an angle θ ˆ is then given by the For the maximum hoop stress criterion the branching angle, θ, angle which satisfies

∂σθ =0 ∂θ

i.e. KI

df1θθ (θ) df θθ (θ) + KII II =0 dθ dθ

θθ (θ) = −3 sin (θ/2) cos2 (θ/2). Therefore, given KI where f1θθ (θ) = cos3 (θ/2) and fII and KII we can solve for θ. The solution for θˆ proved in Section 1.4.8 is given by:   sµ ¶2 1 KI KI ˆ for KII 6= 0. tan (θ/2) =  − + 8 4 KII KII

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).

K I , K II

k I , k II θ

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, kI (θ)2 + kII (θ)2 E0 Problem is to maximize G(θ) which is done numerically. (Note that the usual G G(θ) =

only 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.

ˆ based on three criteria—comparison with experimental Figure 1.20, Branching angle, θ, 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 ˆ then, been determined by the earlier analysis and is designated θ, ³ ´ ˆ = √1 ˆ + KII fII (θ) ˆ σθ (θ) KI fI (θ) 2πr Fracture occurs when the hoop stress reaches a critical value, σc , at a critical distance rc so we get √

´ 1 ³ ˆ + KII fII (θ) ˆ = σc KI fI (θ) 2πrc

ˆ = 1, therefore KIC is the fracture toughness when KII = 0, θˆ = 0, fI (θ) √

2πrc σc = KIC .

Substituting, we get, KI ˆ + KII fII (θ) ˆ =1 fI (θ) KIC KIC 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 (KI fI (θ) + KII fII (θ)) 2πr

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. θ fI (θ) = cos3 ; 2 Criterion is :

fII (θ) = −3 sin

θ θ cos2 2 2

∂σθθ 0 = 0 ⇒ KI fI0 (θ) + KII fII (θ) = 0 ∂θ

ˆ i.e. where 0 denotes differentiation. Denote the solution to this equation to be θ. Ã KI

3 θˆ θˆ − cos2 sin 2 2 2

!

à + KII

" # ! ˆ θˆ θˆ θˆ 3 θ 3 sin cos sin − cos3 =0 2 2 2 2 2

" # ˆ ˆ ˆ ˆ 1 ˆ 3 θ θ θ θ θ ⇒ KI cos2 sin = 3KII cos sin2 − cos2 2 2 2 2 2 2 2 Now sin2

θˆ 1 θˆ 1 ˆ − cos2 = (1 − 3 cos θ) 2 2 2 4

and above becomes KI cos

θˆ 1 θˆ ˆ sin = KII (1 − 3 cos θ) 2 2 2

and since sin 2α = 2 sin α cos α we rewrite this as ˆ KI sin θˆ = KII (1 − 3 cos θ) Thus branching angle θˆ satisfies sin θˆ 1 − 3 cos θˆ Note: for positive KI and KII , θˆ is negative. 27

=

KII KI

For KII = 0 (Mode I), θˆ = 0 for KI = 0 (Mode II), 1 − 3 cos θˆ = 0 ⇒ θˆ = −70.5◦ . (Note that fθθ is negative for θˆ = +70.5◦ , i.e. hoop stress is compressive, see Fig. 3.5 in Appendix C) More general solution for KII 6= 0 sin θˆ 1 − 3 cos θˆ

=

KII KI

KII KII cos θˆ − =0 ⇒ sin θˆ + 3 KI KI Use tan substitution: sin 2α =

2 tan α 1 + tan2 α

cos 2α =

1 − tan2 α 1 + tan2 α

ˆ We get a quadratic in tan (θ/2) i.e. 2t2 −

KI t−1=0 KII

ˆ where t = tan (θ/2). There are two solutions to this equation, the maximum hoop stress is obtained for  ˆ tan (θ/2) =

1  KI − 4 KII

sµ

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 =

1 K2 3π σy2

rp =

1 K2 , π σy2

and for plane stress the result is

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 =

2 1 KIC ¿ a. 3π σy2

In other words a À 0.1

2 KIC . σy2

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

Material behaviour

σy σ

Symmetry boundary

ε

100 90 80 70 60 50 40 30 20 10 0 0

10

20

30

40

50

60

70

80

90

100

Symmetry boundary

1.0

0.75

0.5

0.25

0.0 -1.0

-0.75

-0.5

-0.25

0.0

0.25

crack length = 1

31

0.5

0.75

1.0

Plastic strains PEEQ

VALUE +0.00E+00 +2.00E-03 +4.00E-03 +6.00E-03 +8.00E-03 +1.00E-02 +4.14E+00

2 3

1

Plane strain

Plastic PEEQ strains

VALUE +0.00E+00 +2.00E-03 +4.00E-03 +6.00E-03 +8.00E-03 +1.00E-02 +2.23E+01

2 3

1

Plane stress

Figure 1.23, Crack tip plastic zones for plane stress and plane strain

32

(a)

(b)

(c)

Figure 1.24, Elastic-plastic crack tip fields for plane strain

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 =

R

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 ,

n

Γ

σ, ε

Y

X

Figure 2.1 Closed line contour Γ for a loaded body I ∂u ds, IΓ = W dy − t · ∂x Γ 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 . I IΓ =

W dy − t · Γ

∂u ds ∂x

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

Γ+ Γ1

Γ-

Γ2

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, µ ¶ ∂u IΓ = W dy − t ds = 0. ∂x Γ2 +Γ+ +Γ1 +Γ− Z

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. Z Z Z Z ⇒ = =0⇒ + =0 Γ+

or

Γ−

Γ2

Z

Z

⇒

=− Γ2

Γ1

Z =

Γ1

Γ1−

where the minus sign for Γ1− indicates that the direction of integration is reversed for Γ1 . We define,

Z J=

W dy − t Γ

∂u ds ∂x

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. Z ∂u J= W dy − t ds ∂x Γ 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 ¶ Z π µ ∂u ⇒J = W cos θ − σ · n r dθ. ∂x −π Assume a separable solution for displacement, u, as the crack tip is approached, u = rλ+1 u ˜(θ) ²∼

du ⇒ ² ∼ rλ ²˜(θ) dr

(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 µ ¶ ∂u 1 → W cos θ − σ · n ∂x r

as r → 0

Both terms in the bracket are of order O(σ²) ⇒ σ² →

1 r

as r → 0 39

For a linear elastic material, σ ∼ ². Therefore at the crack tip, σ ∼ ² ∼ rλ . ⇒ σ² ∼ r2λ =

1 r

⇒ 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 (² ∼ σ n ; or σ ∼ ²1/n ) As before, let ² ∼ rλ ²˜(θ) λ

⇒ σ ∼ rnσ ˜ (θ) ⇒ σ² ∼ r Since σ² → ⇒λ=− ⇒

n+1 n λ

.

1 r n n+1

σ ∼ r−1/n+1 σ ˜ (θ) ε ∼ r−n/n+1 ²˜(θ)

This is the HRR singularity, (after Hutchinson, Rice and Rosengren, 1968. For a more general power law material with uniaxial stress-strain law, ² =α ²0

µ

σ σ0

¶n ,

where ²0 , and σ0 are the reference strain and stress and α is a scaling factor, then we can write, σij /σ0 = Ar−1/n+1 σ ˜ij (θ; n);

²ij /α²0 = An r−n/n+1 ²˜ij (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+1 α²0 σ0 In 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 µ σij /σ0 = µ ²ij /α²0 =

J α²0 σ0 In r J α²0 σ0 In r

¶1/(n+1) σ ˜ij (θ; n) ¶n/(n+1) ²˜ij (θ; n).

The terms In , σ ˜ij (θ; n), ²˜ij (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. σ   E µ ¶n ²= σ   ²y σy 41

when σ < σy when σ > σy

Here α = 1, and ²y and σy are used instead of ²0 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), µ uy (r, θ) = α²y

J α²y σy In r

¶n/(n+1) ru ˜y (θ; n) r

θ

∆u Figure 2.6 Definition of crack opening displacement

Thus, for a given material, (i.e. constant n, α, In , ²0 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 = dn 42

J , σy

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.

45

o

δt

o

45

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 nonlinear 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 1

σyy ∼ J n+1 and n

∆u ∼ J 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=−

1 ∂U B ∂a

and for a linear elastic material we have the additional equality J =G=−

K2 1 ∂U = 0. B ∂a 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 elasticplastic 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 K2 Je = 0 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). σy

ML =

W

B

σy BW 2 4

− σy

Figure 2.8, Illustration of limit moment for a plastic beam in bending. Next consider a cracked beam in bending with a << W . 45

B M

M W a

a

Figure 2.9, Edge cracked beam in bending. It can be shown that for plane stress conditions, MLC = σy

B(W − a)2 W 2B = σy (1 − a/W )2 4 4

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

PLC = σy (W − a)B. Limit load solutions are commonly used in fracture mechanics. The ratio between load and limit load is a measure of the extent of plasticity and provides a good means of compar2W

ing two geometries. For example two different geometries at the same ratio of load to limit load have similar J integral 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: ³ a ´2 W 2 B 2 PLC = σy √ 1.22 1 − . W 2L 3 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 = P L/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, 2 2σvm = (σxx − σyy )2 + (σxx − σzz )2 + (σyy − σzz )2

√ ⇒ σvm =

3 σ 2

√ 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: µ 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 =−

1 ∂U . B ∂a

47

Under applied displacement, Z U = Ue = 0

∆

1 P (∆) d∆ ⇒ J = − B

Z

∆ 0

∂P d∆. ∂a

It can be shown that this is equivalent to writing η J= B(W − a)

Z

∆

P d∆ = 0

η A, B(W − 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 = σy

W 2B (1 − a/W )2 2L

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 ∆ = σy

W 2B (1 − a/W )2 ∆. 2L

U = P∆ PL 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, ¯ 1 ∂U ¯¯ 2 W2 J =− = σ (1 − a/W )∆. y B ∂a ¯∆ W 2L Since P = PL = σy

W 2B W2 PL (1 − a/W )2 ⇒ (1 − a/W ) = 2L 2L B(1 − a/W ) ⇒J =

2 A. B(W − a)

i.e. η = 2 for this load configuration. For a center-cracked-tension geometry, , ∆/2

U = P ∆ = σy B(W − a)∆ ¯ 1 1 ∂U ¯¯ = σy ∆ = P∆ J =− ¯ B ∂a ∆ B(W − a) 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

W

the J equation the load per crack tip, i.e. P/2, must be used to have η = 1. , ∆/2

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 2 dC(a) P , 2B da

where C is the elastic compliance P = J=

∆ ∆ ⇒ P2 = P C C

1 ∆ dC 1 1 dC P = A 2B C da B C da

and since the alternative equation for J is J=

η A B(W − a)

⇒ ηe =

W − a dC C 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 ηp Ae + Ap B(W − a) B(W − a) 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 ηp + Ap 0 E B(W − a)

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 A, B(W − 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, ² σ = (E = σ0 /²0 ) ²0 σ0 where σ0 and ²0 are a normalising stress and strain (material parameters). By dimensional analysis we can write σ (x, y) = [σ∞ /σ0 ] f (x, y) σ0 ε (x, y) = [σ∞ /σ0 ] g(x, y) ²0 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

Z J=

W dy − t Γ

∂u ds ∂x

⇒ J ∝ (σ∞ /σ0 )2 σ0 ²0 L 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 = aσ0 ²0

σ∞ σ0 52

¸2 H(a/W )

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 = K 2 /E 0 2 σ∞ aY 2 (a/W ) (in plane stress) E σ 2 ²0 aY 2 (a/W ) = ∞ σ0 ¸2 · σ∞ σ0 ²0 aY 2 (a/W ). = σ0

J=

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, ²/²0 = α(σ/σ0 )n σ (x, y) = [σ∞ /σ0 ]f (x, y, n) σ0 ε (x, y) = [σ∞ /σ0 ]n g(x, y, n) α²0 Note the additional dependence on the hardening exponent n. Using the same argument as before we can write J ∝ ασ0 ²0 (σ∞ /σ0 )n+1 L and thus

·

σ∞ J = ασ0 ²0 a σ0

¸n+1 H(a/W, n).

It proves useful to normalise by the limit load rather than σ0 so we rewrite as ·

P J = ασ0 ²0 a 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 α σ0 ²0 a P/P0 h

[stress][length] dimensionless [stress] dimensionless [length] dimensionless 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

σ   E µ ¶n ²= σ   ²y σy

when σ < σy 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, ²0 = ²y and σ0 = σy and Je = K 2 /E 0 . In order to get good agreement with numerically calculated J values in the elasticplastic 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 nonlinearity. 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, 1 ry = 2π For a small crack, Y =

√

µ

K σy

¶2 .

π (i.e. independent of crack length) and we get √ √ πσ a Keff = s µ ¶2 . 1 σ 1− 2 σy

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 comparison 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 interested 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 elasticplastic 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 = K 2 /E 0 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.

tpb 20 mm

40 mm

ccp

80 mm 800 mm

Figure 2.19, Comparison of K field and numerical crack tip fields for two geometries

58

(a)

(b)

(c)

K = 85 MPa m1/2

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

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, 1 rp = 3π

µ

K σy

¶2

µ = 0.1

K σy

¶2 .

We require a À rp . The ASTM requirements are µ 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 =

p

JIC E 0 .

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. (a)

(b)

(c)

Figure 2.21, Comparison of numerical stress fields and HRR fields for (a) and (b) tpb geometry and (c) ccp geometry

The idea of K and J dominance is summed up by Fig. 2.22, which shows schemat62

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.

ILLUSTRATION OF K DOMINANCE (at low load, or very large specimen) Process Zone

K field HRR field

Specimen

ILLUSTRATION OF J DOMINANCE (at higher load or small specimen) Process Zone

HRR field

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 fracture 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 > 25

JIC . 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 measure as there is usually some stable ductile tearing before final failure. For such materials 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 plotted 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, σf∗ , is given by Gγm σf∗ ≈ p , 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 σf∗ 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 = σf∗ , 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, l0∗ is used here instead of rc ).

σ22

 σ 22  J =  σ 0  αε 0σ 0 I n r 

1 / n +1

σ~22 (0; n)

σ22

Figure 3.2 RKR model for cleavage failure The failure equation becomes, σf∗ = σ0

µ

JIC α²0 σ0 In l0∗

¶1/n+1 σ ˜22 (0; n).

Now, ²0 = σ0 /E, and α and In are fixed for a given material and temperature so we rewrite:

µ σf∗

= Aσ0

EJIC σ02 l0∗

where

µ A=σ ˜22 (0; n) 67

¶1/n+1

1 αIn

, ¶1/n+1

µ ⇒ EJIC =

σf∗ Aσ0

¶n+1 σ02 l0∗ .

As discussed previously, there is a one-to-one relationship between JIC and KIC , 2 i.e. JIC = KIC /E 0 . Then

KIC =

p

µ E0J

IC

=

σf∗ Aσ0

¶(n+1)/2 σ0

p

l0∗ ,

(Taking E 0 = E for simplicity). Rearranging, we get KIC =

(1−n)/2 p ∗ l0 .

1

(σf∗ )(n+1)/2 σ0 (n+1)/2

A

For n = 10 (a typical value for steel), the values are, In = 4.54, σ ˜22 = 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(σf∗ )5 .5σ0−4.5

p l0∗ .

A direct relationship between KIC , the failure stress, the yield strength and the grain size is obtained (since in this model, l0∗ = dg ). Consider the effect of an increase in temperature on the fracture toughness. If temperature is increased, σf∗ is unchanged, σ0 decreases, so KIC increases. If the grain-size is decreased both σ0 and σf∗ increase However, the relative increase in σf∗ 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)

l0* 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∗f = ε0

µ

JIC α²0 σ0 In l0∗

¶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∗f (ασ0 In l0∗ ). ε˜

Writing in terms of KIC we get, r KIC =

αIn q σ0 E 0 In ε∗f l0∗ . ε˜

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 (1.5σm /σy )ε˙∞ p ,

69

∞ where r is the void radius, r˙ is the rate of increase of the radius, σm 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 associated 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 εf 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)). (a) Dp

dp initial crack (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 (1.5σm /σy )ε˙∞ p ,

we can integrate from the initial void size (or void initiating particle), Dp , to the final 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,

Z

dp /2

⇒ Dp /2

dr = 0.558 sinh (1.5σm /σy )dεp r dr = 0.558 sinh (1.5σm /σy )ε∗f , r 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 Bridgeman equation gives an approximate solution for the triaxiality in the notch as µ ¶ σm 1 amin . = + ln 1 + σe 3 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.

3.0 sharp crack

σm σ

uniaxial

0.3 0.2

1.2

ε *f

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).

K IC =

αI n σ 0 E ′ε f*l0* ~ ε

&WEVKNG

KIC $TKVVNG 6TCPUKVK




VGORGTCVWTG

6GORGTCVWTG

K IC =

1 A

( n +1 )/ 2

(σ ) * f

(1+ n )/ 2

σ 0(1−n )/2 l0*

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 structures. 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, K KIC δ δr = δc σref P Lr = = σy PL σref P Sr = . = σf PL (σf )

Kr =

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., √ and

s J K =p , σy Eσy

δ∝ s

p

δc ∝

Hence

r

JIC KIC =p . σy Eσy δ K = , δc 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 F (σ/σy ), σy E

where F (σ/σy ) is an adjustment factor which accounts for plasticity effects on the CTOD. It is given by the following expression, 1   ¶ F = µ σ ¶−2 µ σ   − 0.25 σy σy

for σ ≤ 0.5σy 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 subdivided 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 = ασy ²y a(P/PL )n+1 h(a/w; n), where PL is the limit load, and the stress-strain relationship of the material is given by ²/²y = 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, ²ref = α²y (σref /σy )n , we can rearrange the J equation above to get J = a²ref σref h(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 = or inverting

Kc2 , E0

E0 1 ≤ . 2 Kc J

Multiplying across by Je = K 2 /E 0 we get failure when µ

K Kc

¶2 =

Je , J

where K is simply the linear elastic stress intensity factor. We can rewrite in terms of Kr , K Kr = = Kc

µ

J(σref ) Je

¶−1/2 .

In other words, the curve for the FAD can be determined directly from the relationship 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 2 σref h(a/W ; 1), E since for a linear elastic material, α = 1 and ²ref = σref /E. Then we have

Je = a

J E²ref = , Je σref making the assumption that h(a/W ; n) = h(a/W ; 1), and then µ Kr =

J Je

¶−1/2

µ =

79

E²ref σref

¶−1/2 (∗).

Since Lr = σref /σy and ²ref 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, √ √ πσ a Keff = s µ ¶2 . 1 σ 1− 2 σy 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

µ

Keff K

¶2 =

1 1− 2

1 µ

σref σy

¶2 .

and using the binomial theorem for small σref /σy we get J 1 =1+ Je 2 80

µ

σref σy

¶2 (#).

2 Note that we have used the equation J = Keff /E 0 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: E²ref 1 J = + Je σref 2

µ

σref σy

¶2 µ

E²ref σref

¶−1 .

The term E²ref /σref appears in both of the terms on the RHS of the equation. For σref < σy , ²ref = σref /E so E²ref /σref = 1 and the small scale yielding equation is obtained. Under large scale plasticity it may be seen that E²ref /σref ≈ ²pl /²el 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, µ Kr =

J Je

¶−1/2

µ =

E²ref 1 L3r σy + Lr σy 2 E²ref

¶−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 p

¡ ¢ δr or Kr = (1 − 0.14L2r ) 0.3 + 0.7 exp (−0.65L6r )

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 essentially 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 µ 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 temperatures 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 characterised by a power law creep relationship, analogous to power law plasticity, ²˙c = ²˙0

µ

σ σ0

84

¶n ,

where n, σ0 and ²˙0 are material constants. This equation is identical to the power law plasticity relationship but with ²˙0 replacing ²0 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. µ σij /σ0 = µ ²˙cij /²˙0

=

C∗ ²˙0 σ0 In r

C∗ ²˙0 σ0 In r

¶1/(n+1) σ ˜ij (θ; n)

¶n/(n+1) ²˜ij (θ; n).

The parameter C ∗ is the creep analogy to J and is defined in the same way as J using a contour integral,

Z ∗

C =

∂ u˙ ˙ (ε)dy ˙ W − t ds, ∂x Γ

where ˙ (ε) ˙ = W

Z ε˙ σdε˙ 0

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 independence 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 A, B(W − a)

where A is the area under the load displacement-rate curve. Note that analogous to pure power law plasticity, A=

n ˙ P ∆, n+1

˙ is the remote displacement rate corresponding to the load P , allowing the where ∆ above equation to be simplified to 85

C∗ =

η n ˙ P ∆. B(W − a) n + 1

Values of η for various crack geometries are available in the fracture mechanics handbooks 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 ∗ , µ ∗

C = a²˙0 σ0

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, E²ref J = , Je σref with σref defined via the plastic collapse load. Similarly for power law creep, we can write, C∗ E ²˙ref = . Je σref It is convenient in the creep case to replace Je by K and the equation becomes C∗ =

K 2 ²˙ref , σref

and we then write C ∗ = σref ²˙ref R0 , where R0 is a length scale which depends only on the geometry of the component, R0 = (K/σref )2 , e.g. for a crack in an infinite plate R0 = 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 comparable. 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, µ

¶1/(n+1) C(t) σij /σ0 = σ ˜ij (θ; n) ²˙0 σ0 In r µ ¶n/(n+1) C(t) c ²˙ij /²˙0 = ²˜ij (θ; n), ²˙0 σ0 In r 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

Z

∂ u˙ ˙ (ε)dy ˙ W − t ds. ∂x Γ→0 Figure 5.3 shows the variation of C(t) with time determined from a numerical C(t) =

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 K2 . (n + 1)E 0 C ∗ For the case illustrated in Fig. 5.3 it may be seen from the numerical result, that tT =

the time to reach steady state tss ≈ 3tT . The approximate solution to C(t) shown in the figure is given by the equation, µ C(t) =

tT t

¶ C∗ =

K2 , (n + 1)E 0 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 applicable 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 stationary 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 R0 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 cur89

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 ²f 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, µ ²˙cij

= ²˙0

C∗ ²˙0 σ0 In r

¶n/n+1) ²˜ij (θ; n),

the creep strain accumulated over a time t is given by Z c

² =

µ

t

²˙c dt = ²˙0 ²˜(0; n) 0

C∗ ²˙0 σ0 In

¶n/n+1 Z

t 0

1 r(t)n/n+1

dt,

where ²c 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, µ c

² = ²˙0 ²˜(0; n) Since

C∗ ²˙0 σ0 In

¶n/(n+1) Z

0 rc

dr 1 dt = 1/ =− , dr dt a˙ 90

1 rn/n+1

dt dr. dr

we can write

¶n/(n+1) Z rc C∗ 1 dr n/n+1 ²˙0 σ0 In r 0 µ ¶n/(n+1) ²˙0 ²˜(0; n) C∗ rc1/(n+1) . = (n + 1) a˙ ²˙0 σ0 In

²˙0 ²˜(0; n) ² = a˙ c

µ

Crack growth occurs when this strain is equal to the material ductility, ²f . 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, ˙ ²˙0 ²˜(0; n) a˙ = (n + 1) ²f

µ

C∗ ²˙0 σ0 In

¶n/(n+1) rc1/(n+1) .

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, ²f . The triaxiality, σ ˜m /˜ σe , under plane strain conditions is about a factor of three higher than plane stress, leading to a creep failure strain ²f 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 . ²f

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, ²f 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 Z c

² =

t

²˙c dt = t²˙c .

0

Note that the above assumes that even during incubation, most of the strain accumulated 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, µ c

² = t²˙0 ²˜(0; n)

C∗ ²˙0 σ0 In rc

¶n/n+1

and ²f ti = ²˙0 ²˜

µ

²˙0 σ0 In rc 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:

·

σref tr ti = 0.0025 K2

¸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 fracture 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 distribution are closer to the creep distributions than the elastic ones. The equations derived previously still hold with the requirement that for steady state, ²˙c > ²˙e + ²˙p , where ²˙e and ²˙p 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 mechanics “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 toughness 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 Mechanics (Appendix B and C may be used freely in exams) Strain energy density, W, for a linear elastic material under uniaxial stress, σ : W=

σ2 2E

Strain energy density, W, for a power law material under uniaxial stress, σ : W=

n n+1

σε

Definition of strain energy Ue: U e = ∫ WdV V

Strain energy for a linear elastic material under an applied force, P, giving rise to a displacement, ∆ : P∆

Ue =

2

Strain energy for a linear elastic material under an applied moment, M, giving rise to a curvature, θ : Mθ

Ue =

2

Strain energy for a power law material under an applied force, P, giving rise to a displacement, ∆: n

Ue =

n +1

P∆

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, G=−

1 δU B δa

Energy release rate-compliance relation for a linear elastic material: G=

1

P2

dC

2B

da

Definition of K for a cracked geometry: K = Yσ a ; Center crack in an infinite plate under tension, Y = √π ; Edge crack in an infinite plate under tension, Y = 1.12√π. Page 1 of 4

97

Crack opening displacement for a Mode I crack in a linear elastic material:

∆u ( r ) = 8

K

r

; E ′ = E for plane stress; E ′ =

E ′ 2π

E 1 −υ2

for plane strain.

Relationship between energy release rate and stress intensity factor: 2 K I2 + K II2 K III + ; G = shear modulus 2G E′

G=

K II

Phase angle, φ = tan -1

KI

; Mode mixity, M =

2

π

φ.

Branching angle, θ, for a crack under mixed mode conditions in a linear elastic material, using the maximum hoop stress criterion:  

2

 



1 KI − 4 K II

tan (θ 2 ) =

KI 












K II



+8

θ

,


Plastic zone size, rp, ahead of a Mode I crack: 2

 

K

rp = α







σy 

; α=

1

π

for plane stress; α =

rp

1 for plane strain 3π

Plastic zone correction for K: K = Y (a e )σ a e ; a e = a + rp / 2 ASTM size requirement for KIC testing: a, b, B > 2.5(KIC/σy)2. Definition of J line integral: Γ J = Wdy − t

Γ

δu δx

y

ds

x

HRR crack tip stress and strain distribution for a power law hardening material:

σ ij J = σ0 αε 0σ 0 I n r





1 /( n +1)

ε J σ~ij (θ ; n ); ij = αε 0 αε 0σ 0 I n r



n /( n +1)











ε~ij (θ ; n )

Relationship between crack opening displacement, δ, and J integral:

δ = dn

J

σ0

; d n ≈ 1 for plane stress; d n ≈ 0.5 for plane strain.

Page 2 of 4

98

Plane stress limit moment for a shallow cracked beam in bending:

B (W − a )2

ML =σ y

4

Plane stress limit load for a shallow cracked beam in tension: PL = σ y B (W − a )

Evaluation of J using the η factor:

J=

K2 E′

+

η B (W − a )

A p ; η ≈ 1 for tension loading; η ≈ 2 for bend loading.

Evaluation of J using GE-EPRI equation: n +1 

J=

K2 E′

+ ασ 0ε 0 a

P 






P0

h (a / W ; n )

ASTM size requirement for JIC testing (for deep cracked bend specimen)

a, b, B > 25(JIC/σy) Rice and Tracey void growth relation:

r = 0.558sinh (1.5σ m / σ y )ε ∞p r 



BS7910 Failure Assessment Diagram: Kr =

σ σ K P P ; Lr = ref = ; S r = ref = K IC σ y PL σf PL (σ f )

P

σ ref =











PL

σ y ; σ f = 1.2σ y or (σ y + σ u ) / 2 −1 / 2



J

Failure condition: K r =











Je

(

)(

(

BS7910 Level 2A FAD, K r = 1 − 0.14 L2r 0.3 + 0.7 exp − 0.65 L6r −1 / 2

 

Eε ref 

BS7910 Level 2B FAD, K r = 



Lrσ y

+

3 1 Lrσ y



2 Eε ref 

Page 3 of 4

99

))

Lr max =

σf σy

Definition of C* line integral:

δu C* = ∫ W dy − t ds δx Γ Evaluation of C* for a power law material using the η factor: C* =

η

n

B (W − a ) n + 1

P∆

Evaluation of C* using GE-EPRI equation:  P C* = aε0σ 0  P  0

   

n +1

h (a / W ; n )

Equation for C* based on σref:  K C* = σ ref εref R′ ; R′ =   σ ref

   

2

Creep crack growth rate equation: a = A(C * )φ

Page 4 of 4

100

6.3 Appendix C, Linear Elastic K field distributions (Appendix B and C may be used freely in exams)

101

102

103