Griffith's Analysis + Fatigue + Fracture

8/2/2019 Griffith's Analysis + Fatigue + Fracture

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Natural Sciences Tripos Part II

MATERIALS SCIENCE

C15: Fracture, Fatigue and Creep Deformation

Dr C. Rae

Michaelmas Term 2010-11

II


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Part II Materials C15 Michaelmas 2010 1

C15: FRACTURE FATIGUE AND CREEP DEFORMATION

Catherine Rae 12 LecturesSynopsis

Introduction: This course examines the use of fracture mechanics in theprediction of mechanical failure. We explore the range of macroscopic failuremodes; brittle and ductile behaviour. We take a closer look at fast fracture inbrittle and ductile materials characteristics of fracture surfaces; inter-granularand intra-granular failure, cleavage and micro-ductility. We describe the range offatigue failure and apply fracture mechanics to the growth of fatigue cracks.Finally we look at the processes of creep and how it combines with fatigue.

Griffiths analysis: Revision of concept of energy release rate, G, and fractureenergy, R. Obreimoffs experiment. Timeline for developments.

Linear Elastic Fracture Mechanics, (LEFM). We look at the three loading

modes and hence the state of stress ahead of the crack tip. This leads to thedefinition of the stress concentration factor, stress intensity factor and thematerial parameter the critical stress intensity factor.

Superposition principle, prediction of crack growth direction, failure of thinfilms.

Plasticity at the crack tip and the principles behind the approximate derivationof plastic zone shape and size. Limits on the applicability of LEFM. The effect ofConstraint, definition of plane stress and plane strain and the effect of component

thickness.

Concept of G - R curves, measuring G and K.

Elastic-Plastic Fracture Mechanics; (EPFM). The definition of alternativefailure prediction parameters, Crack Tip Opening Displacement, and the Jintegral. Measurement of parameters and examples of use.

The effect of Microstructure on fracture mechanism and path, cleavage andductile failure, factors improving toughness,

Fatigue: definition of terms used to describe fatigue cycles, High Cycle Fatigue,Low Cycle Fatigue, mean stress R ratio, strain and load control. S-N curves.

Adapting data to real conditions: Goodmans rule and Miners rule. Micro-

mechanisms of fatigue damage, fatigue limits and initiation and propagationcontrol, leading to a consideration of factors enhancing fatigue resistance.

Total life and damage tolerant approaches to life prediction, Paris law.

Creep deformation: the evolution of creep damage, primary, secondary andtertiary creep. The use of Larson-Miller parameters. Micro-mechanisms of creep

in materials and the role of diffusion.

Ashby creep deformation maps. Stress dependence of creep power lawdependence. Comparison of creep performance under different conditions extrapolation and. Creep-fatigue interactions.


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Booklist:

T.L. Anderson, Fracture Mechanics Fundamentals and Applications, 2nd Ed. CRCpress, (1995) (Fracture mechanics and its application to fatigue, very thoroughand readable)

B. Lawn, Fracture of Brittle Solids, Cambridge Solid State Science Series 2nd ed1993. (Exactly as it says on the label very good on LEFM)

J.F. Knott, P Withey, Worked examples in Fracture Mechanics, Institute ofMaterials. (Excellent short summary of fracture mechanics and good workedexamples)

H.L. Ewald and R.J.H. Wanhill Fracture Mechanics, Edward Arnold, (1984).(Provides very clear explanations different perspective from Anderson)

S. Suresh, Fatigue of Materials, Cambridge University Press, (1998)

(Excellent on fatigue but not very readable)

L.B. Freund and S. Suresh, Thin Film Materials Cambridge University Press,(2003) Chapter 4 for a very thorough description of failure of thin films (evenless readable than the above).

G. E. Dieter, Mechanical Metallurgy, McGraw Hill, (1988)

(Good entry-level text on mechanical properties)

D.C. Stouffer and L.T. Dame, Inelastic Deformation of Metals, Wiley (1996)

(Particularly chapters 2 and 3 for creep and fatigue)

R.C Reed, The Superalloys, CUP (2006). Particularly Chapters 2 and 3 for creepand fatigue in superalloys and Chapter 4 for lifing strategies.


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FRACTURE, FATIGUE AND CREEP DEFORMATION

SYNOPSIS

This course examines the use of fracture mechanics in the prediction of

mechanical failure. We explore macroscopic failure modes; brittle and ductilebehaviour, and take a closer look at fast fracture in brittle and ductile materials characteristics of fracture surfaces; inter-granular and intra-granular failure,cleavage and micro-ductility.

Fatigue causes 90% of engineering failures: we examine how we characterise thesusceptibility of materials to fatigue and estimate lifetimes.

At high temperatures time-dependent plastic deformation occurs: we describe themechanisms of creep and how it can both exacerbate and mitigate the effects offatigue.

GRIFFITHS THEORY, REVISION FROM 1B COURSE.

Griffiths Theory provides the thermodynamic or energetic criterion for failure: itdoes not consider the mechanism by which failure occurs.

The basic premise is that a crack will propagate in a material when the elasticenergy released as a result of that propagation exceeds the energy required topropagate the crack. In the first instance just the surface energy needed to

create two new surfaces was considered, but this applies only to ideal brittlesolids i.e. those where fracture occurs without any plastic deformation.Subsequently this was widened to include the work required to perform theplastic deformation associated with ductile failure and, in principle, can include

any work necessary such as de-cohesion on composites phase changes etc.

2a

If we introduce a crack of length 2a into an infinite plate of thickness B under a

uniform stress , the elastic stresses relax around the crack and reduce the

elastic potential energy UE stored in the plate. Extra surface is created at thecrack, US, and, if the grips are fixed, no external work, UF, is done by the appliedforce, UF = 0.


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sEF UUUaU At equilibrium:

dU

da

dUE

da

dUS

da

0

The change in the potential energy is estimated from an elastic analysis of the

stresses around the crack:

UE 2a2B

E

And the work done to propagate the crack is:

sS aB4U

Where the area of the crack is 2aB, the surface area is 4aB and the surface

energy is s.

Thus:

d(UE)

da2B2a

Eand

dUSda

4Bs :

hence:E

a2

2

s

Rearranging:a

E2 s

Griffiths Equation

This is for an ideal brittle solid; for a ductile material the plastic work of

deformation p , is introduced:

a

E)2( ps

Modification of the fracture criterion to include plastic work leads to the moregeneral definition of the energy release rate or the crack extension force: G.This is the change in the potential energy, U, of the system per unit increase incrack area, A, and has the dimensions of force/length.

Energy Release Rate:

G dU

dA

dU

2Bda

2a

E

According to Griffiths crack extension occurs when this equals the work tofracture, 2s + p .

psc 2GG Gc is a material constant and a measure of the fracture toughness.

The rhs is the resistance to crack growth is termed R where R = 2s + p.


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OBREIMOFFS EXPERIMENT

A real example illustrates two important points: firstly that brittle fracture is

reversible under the right circumstances and secondly, that whether it occurs ornot is governed by balancing stored elastic energy with the work of fracture.

In 1930 Obreimoff split a thin sheet of mica off a larger piece by inserting a

wedge of thickness h beween the layers. The crystal cleaves along the weakinterfaces between the layers to give a thin upper fillet and a thick lower section.As the wedge is driven into the crack the crack grows to keep the lengthconstant. The elastic energy stored as the wedge is forced into the open crack isprincipally in the thin upper fillet, and is balanced by the cohesive forces at thecrack tip. The crack opens until these are balanced. The energy is calculatedeasily from the elastic properties of the mica, and the geometry of the set-up.

The elastic strain in the cantilever is given by beam theory:

U UE Ed3h2

8a3where the constants are given in the diagram.

The surface energy needed to grow the crack is

US 2a where is the surface energy.Equating the elastic energy to the surface energy gives an equilibrium cracklength ao of:

ao 3Ed3h2 /164

As the wedge is withdrawn the crack closes and the damage is pretty muchrepaired if the process is done in vacuum. This can be shown by reopening the

crack and noting that the value of ao for the re-opened crack is almost the same.As air and moisture are introduced, the quality of the repair deteriorates and theequilibrium length ao increases.


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TIME LINE

Fatigue Fracture

~1500 - Leonardo da Vinci failure stress of iron wiresdepends on length i.e. on probability of flaw

1842 - Railway accident Versailles - failure of axle

1843 - significance of fatigue striations recognizedWJM Rankin

1852-1869 - Wohler systematic experiments on bendingand torsion development of S-N curves

1874 & 1899 Gerber and Goodman life predictionmethodologies

1886 Baushinger effect noted

1900 Ewing and Rosenberg recognition of persistent

slip bands extrusions and intrusions

1913 Inglis elastic stress field around elliptical hole

1920 Griffiths equation for brittle materials

1930

1938

Obreimoffs experiment

Westergaarde elastic solution of the stressdistribution at a sharp crack

1945 Constance Tipper and the Liberty ships -

Recognition of the Ductile Brittle transitionTipper test and the role of crystal structure infailure

1945 Minor accumulation of fatigue damage

1953 -54 Comet airliner losses due to fatigue failure

1954 Coffin Manson empirical laws for HCF and LCF

1956 1956 Wells applies fracture mechanics to fatigue to

explain the Comet fatigue fractures

1956 Irwin development of the concept of energyrelease rate based on Westergardes work

1956 Demonstration of the role of PSB in initiating

fatigue failure

1957 Fracture mechanics predicts disc failures for GE

1960 1960 Paris law relating the crack growth rate to the

stress intensity factor

1960-61 Irwin/Dugdale/Wells development of LEFM andeffect of plastic zone size and shape

1968 Proposal of the J integral by rice and the CTOD

by Wells to cope with the failure of ductilematerials

1976 Shih and Hutchinson establish the theoretical

basis of the J-Integral and link it to the CTOD

1980 Chaboche Development of time dependantfracture interactions between creep andfatigue.


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WHAT IS A BRITTLE FRACTURE?

Brittle

Brittle

Ductile

Ductile

Very few fractures are truly brittle i.e. have no permanent deformation.

But fracture is still determined by the energy balance and the energy driving thecracking process is still the elastic energy stored in the cracked body. Fastfracture is a more accurate term than brittle fracture to use for rapid failure.

Where local deformation occurs the cracking process is not reversible as it was inthe case of Mica.

Can deal with a great many materials and situations using simple elasticassumptions. This is known as linear elastic fracture mechanics.

[There is a fundamental flaw inherent in LEFM the calculations assume elastic

behaviour but we know that for the crack to have any chance of growing thestresses at the tip must vastly exceed the yield stress: yet we carry on anyway!The point is that in many materials the contribution to the energy balance from

the non-elastic part is a tiny fraction of the total equation. We can put this to oneside for the time being, but will examine this later.]


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LINEAR ELASTIC FRACTURE MECHANICS

When a crack occurs in a material the local stress around the crack is raised.LEFM relies on the sufficient of the specimen/component being elastic such thatthe energy release rate can be calculated from the elastic displacements around

the crack tip. Hence if you can solve for the elastic stress in any configurationyou can (in principle) calculate G from dUE/da.

STRESS CONCENTRATION AT FEATURES

In some simple situations the equations governing elastic deformation can besolved analytically:

i. Expressing the stresses in terms of complex potentialsii. Specifying the boundary conditions

iii. Finding functions to satisfy the above

Or, more generally, solving the problem using finite element analysis. Oneproblem for which there is a solution is that of a circular hole in an infinite thinplate subject to a stress o.

In polar co-ordinates the stresses

are given by:

rr

o

21

ro2

r2

1 3ro4

r4

4ro2

r2

cos2

o2

1 ro2

r2

1 3ro4

r4

cos2

r o2

1 3ro4

r4

2ro2

r2

sin2

Substituting r = ro and = 90 and 0: gives the maximum and minimum hoopstresses , at the edge of the notch as 3o and -o. Thus the presence of a

round hole in the plate increases the tensile stress by a factor of three in one

direction and introduces a compressive stress at the top of the hole equal to thedistant tensile stress.


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Because all the stresses are elastic and therefore small, the imposed stress fields,and the solutions for those stress fields, can be added: this is known as thePRINCIPLE OF SUPERPOSITION.

Hence, adding two stresses o at right angles to each other to produce a 2Dhydrostatic tension and the stresses around the hole in the plate are now:

3o- o = 2o.

Another important situation for which an exact solution exists is that of anelliptical hole, semi-axes a and b, in a plate, subject to a distant stress o. In this

case the maximum stress is at the tip of the ellipse:

2a

2b2

o

max o 12a

b

or

max o 12a

wherea

b2 the radius tangential at the tip.

Hence for a long thin crack where a >>b,

max o 2a

This is slightly modified for a half crack at the edge of a plate by the factor 1.12because the free surface (zero stress) allows the ellipse to open rather wider than

for the embedded crack.

The factor max/o by which the elastic stress is raised by a feature such as a

crack or a hole is the stress concentration factor kt. This is dimensionless.


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SHARP CRACKS

The above is very useful for finding the effect of features (intended or

unintended) in the structure, but most cracks are long and have sharp tips.These can be of atomic dimensions in brittle materials.

In 1939 Westergaard solved the stress field for an infinitely sharp crack in aninfinite plate. The elastic stresses were given by the equations;

yy o a

2rcos

2

1sin

2

sin

3

2

xx

o a

2r cos

2

1sin

2

sin3

2

xy o a

2rsin

2

cos

2

cos

3

2

+ similar expressions for displacements u

[Equations for the polar stresses as a function of r and are in the data-book.]

All the equations separate into a geometrical factor and the stress intensityfactor:

K o a

K determines the amplitude of the additional stress due to the crack over the

whole specimen, but particularly at the crack tip where growth has to occur.

When = 0 the stress opening the crack has the value :

yy o a

2r

K

2r

The value of K at which fracture occurs is the material-dependant

Fracture Toughness:

KIc f a

For a fixed stress this defines the maximum stable crack length or for a fixedcrack length the maximum stress.

r


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You have come across K in 1A and 1B: Be careful, there are a number ofparameters K:

kt max

ostress concentration factor (dimensionless)

K a stress intensity factor Pa m

KIc f a critical stress intensity factor Pa m or Fracture Toughness

The equations indicate an infinite stress at the crack tip when r = 0. This is not a

problem as the stored elastic energy forms a finite interval. A small volume atthe crack tip will be above the yield stress and thus in a plastic state.

OTHER MODES OF FAILURE PRINCIPLE OF SUPERPOSITION

The above equations considered only a stress normal to the crack surface butmuch more complex states of stress will exist at cracks. These can be resolved in

to three distinct crack opening modes, termed with extraordinary imagination,modes I II and III. Combinations of these can describe any state of stress andthe stresses are additive as they remain elastic.

For example the mode II stress equations include the factor

KII r , for anylocation around the crack tip since the stresses are additive, the values of K fromthe separate crack modes are also additive.

Crack opening modes I, II and III.

The energy release rate is given by integrating stress strain with respect to r,and has the value:

GK2

EFor plain stress, or

GK2

E(1 2) for plane strain.

Hence, because the values of K for each opening mode can be assessedindependently and then added, it is possible to assess complex multimodecracking modes.

The total change in energy in the body as a whole can be expressed directly interms of the individual stress intensities which characterise the crack tip stressand displacement fields. The total energy release rate is given by the expression:


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EGKI2 KII

2 (1 )KIII2

For plane stress or:

EG (1 2)KI2 (12)KII

2 (1)KIII2

For plane strain.

Note: These equations do not include the background stress which must be

added.

ys

o

K dominated

Overall stress

r

Plastic zone

Diagram showing the net stress resulting from the remote stress and the stressintensity . For o


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The stress on the crack inclined at 45 can be resolved into a component actingperpendicular to the crack, i.e. in Mode I, and a component acting parallel to thecrack plane, i.e. in Mode II. The stress fields from each can be considered

separately and then combined to give the overall energy release rate for the newcrack. The path the crack takes in propagating further will be that which

maximizes the total energy released. We can find this by differentiating theenergy release rate with respect to the angle .

The radial stresses for Mode I and Mode II growth are given below using polar co-ordinates because we will be looking to define the angle of propagation of the

crack. (Note the trig. expressions below are given in the data book)

Mode I:

rr

r

KI

2r 1 2cos(/2)[1 sin2(/2)]

cos3(/2)

sin(/2)cos2(/2)

Mode II:

rr

r

KII

2r 1 2sin(/2)[13sin2(/2)]

3sin(/2)cos2(/2)

cos(/2)[13sin2(/2)]

The axes for the above equations are located in line with the existing crack. Wehave two independent stress fields from the mode I and II stresses on this crack.We use these stresses to work out what the energy release rate for a small

(virtual) crack taking off at an angle from the end of the main crack. For the

crack continuing in the same direction would be zero etc, see diagram above.

We extract the stresses which will cause mode I opening of the virtual crack;these are the values from each of the stress fields.

From perpendicular stress:

o

2

a

2rcos3

2

where the factor 1/2 = cos

From parallel stress:

o

2

a

2r3sin

2cos2

2

Total stress opening crack in mode I by adding the above:

o

2

a

2rcos2

2cos

23sin

2

Similarly the stress to cause mode II opening comes from the r components:

r o

2

a

2rsin

2cos2

2 cos

213sin2

2


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These stresses convert into K values: KI()= a and KII() = ra

The K relates to the very small new crack growing at the end of the main crack.

We now have to find the value of for which the energy release rate will be amaximum, and do this by adding the G values for each of the two modes of

opening:

G()=(KI)2/E +(KII)

2/E = 2a/E + r

2a/E

We are only concerned with the angle and can plot the normalised contributions

for Mode I and Mode II opening combined, normalised by the values at = 0

Plot of normalised energy release rate for propagation of a crack angles at 45 tothe principal stress direction.

These are plotted above, and it can be seen that the mode I crack opening modehas a very strong maximum at ~-55 corresponding to a minimum in the Mode IIcrack. Nevertheless, the sum of the two, denoted by the bold line, is dominated

by the energy released from Mode I (as is nearly always the case).

It should be stressed that K still remains a: the inclusion of the angularfunction in calculating K is a result of using the stress field from the main crack to

generate the energy release rate of the new crack going off at an angle .

This illustrates how the principle of superposition works both Mode I and ModeII cracks could grow given sufficient stress. The KIC and KIIC values for aparticular material are different and characteristic of that material. In practicenearly all cracks grow in Mode I this normally generating the highest energy

release rate.


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FRACTURE OF THIN FILMS

Fracture mechanics is increasingly being employed to very small scale problems

such as the spallation of thin films or the delamination of layered structures. Thisis relevant to issues such as oxidation, coatings, composites etc.

Generally a film bonded to a substrate will have a different thermal expansion

coefficient, with the result that there is a residual stress in the coating due to thecoating process, or that one develops during thermal cycling. Assuming that themisfit between the two does not exceed the yield strain of the material and thatthe film remains bonded to the substrate, this can be accommodated elastically.Far from the free edges the film stress is constant and in two-dimensional planestrain i.e. there is no traction force between the film and the substrate.

But either at the edge of the film or if the film breaks creating an edge, a shearstress will develop between the film and substrate which transfers the biaxialstress from the film to the substrate so that the stress in the film diminishes tozero at the free edge. This gives rise to a stress state where a Mode II type crackcould propagate along the (weak) interface.

For points close to the edge of the film (relative to the thickness of the film df)

there is also Mode I stress as the edge of the film curls up in response thevarying tension through the thickness of the film. These forces gradually diminishas the distance from the film edge increases and reach about 90% of the remotevalue in a distance comparable to the film thickness (this depends on the relativeelastic properties etc see Suresh: Thin film Materials Ch. 4).

For a film delaminating from a substrate at steady state the energy release rate is

proportional to the film thickness df, and the remote misfit stress squared, m2:

G1 f

2

2Efm

2df where

m Ef

1 f mfor plane strain

Note that the crack length a does not matter if the stress in the film is relaxed asthe film delaminates. However, if some stress is retained, for example by acontinuous film maintaining the stress or a curved front, the driving force will

eventually drop below that needed for propagation of the crack and it will stop.


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In most realistic problems not only is the geometry the elastic propertiesof the two materials differ and this greatly complicates calculating the sizeand distributions of the stresses. Such problems are best addressed using

finite element modeling.

The mode I stresses can make cut edges particularly vulnerable to spallation butthe relaxation of the substrate can also reduce the value of G.

Breaks or cuts in the film surface lead to a greater stress as the substrate cannotstrain locally to reduce the strain in the film.

If the film delaminates but remains intact there will in principle be no tractionforces at the crack edges unless and until the film actually breaks.

If the film is in compression during some part of the cycle there is a risk of

buckling and thus breaking (this typically occurs in protective oxide films).


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PLASTIC ZONE SIZE

The equations above indicate an infinite stress at the crack tip when r=0. Thus a

small volume at the crack tip will be above the yield stress and thus in a plasticstate. This has two effects:

1. The deformation occurring in the plastic zone as the crack grows greatlyincreases R, the work to propagate the crack.

2. The nominal elastic energy stored in the plastic zone is not released as the

crack grows, but, provided the plastic zone remains small, this is a smallproportion of the integral evaluating the energy release rate. Hence, forsmall plastic zone size linear elastic fracture mechanics can be applied toductile failure.

How big does the plastic zone size need to be before we need to modify theenergy release rate equation? This occurs when the elastic energy not stored in

the plastic zone represents a sizeable proportion of the total energy release rateG. Calculating the plastic zone size is not easy, and we rely on a couple ofapproximations (Dugdale and Irwin, see Ewalds page 56) to estimate the effect.They give similar results and so we will look briefly at only one method, that due

to Irwin.

The simplest estimate is made by assuming that the area ahead of the crack tip

where the stress exceeds the yield stress is plastic; (see previous diagram). Thusignoring the remote stress, the size of the plastic zone rp is:

ys KI

2ry

hence

ry 1

2

KI

ys

2

This, however, takes no account of the redistribution of the stress which wouldhave been carried by the material at the crack tip which has yielded and can onlycarry the yield stress.

We can estimate the error by assuming a plastic zone, width 2ry ahead of thecrack tip. The effect of the plastic flow is to open the crack more widely than thepurely elastic response would predict, thus the elastic field of the crack behaves

as if it were a longer than it really is. The tip of the virtual crack acts as thenominal centre for the stress and strain fields resulting from the crack and for

the associated plastic zone.

Diagram showing elasticstress redistribution as aresult of yielding Irwinmodel.

A


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The extent of the extended plastic zone is defined by the yield stress.

ry 1

2

KIys

2

2

2ys2

a a

Where

KI a a and the new plastic zone size is:

rp a ry

Irwin determined a on the basis that the average of the nominal stress in theplastic zone in the plane perpendicular to the stress axis should equal the real

stress, i.e. the yield stress. Then the load is being supported by the crackedcomponent remains the same with and without the plastic zone. In effect thearea under the stress graph, A, is set equal to ysa.

ysa a a

2rdryry

0

ry

ys a ry a a

2rdr

0

ry

ys a ry 2 a a2

ry but

ys 2ry a a from above

ys a ry 2ys 2ry

2ry

a ry and

a1

2

KIys

2

and

rp 1

KIys

2

2ry

Thus the virtual crack tip determining the elastic stress/strain field ends at thecentre of the plastic zone.

Dugdales analysis is rather more sophisticated but also assumes that the crack islonger than it really is and superimposes point closure forces onto each end of the

crack onto the overall elastic solution for the enlarged crack. The criterion for theimposed closure stress is that the sum of the closure and remote stresses cancel

at the crack tip removing the singularity. (see Anderson page 77)

Dugdales analysis gives a slightly larger plastic zone size:

rp 0.392KI

ys

2

instead of

rp 0.318KI

ys

2

from Irwin.


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It is not worth worrying too much about these factors as both analysis arepredicated on perfect plastic behavior, i.e no work hardening. In fact materialswill work harden to different extents and would thus be able to sustain higher

loads in the plastic zone than these analyses predict. FE analysis provides abetter method of assessing the plastic zone size for each material from its

particular plasticity characteristics.

REAL PLASTIC ZONE SIZES

We can use this to estimate the error introduced by the plasticity at variousratios of the stress to the yield stress.

ys MPa KIC MPam1/2 ASM

Crit.

rpplane

stress

rp Plane

strain

High strength Steel 1200 60

Structural steel 400 150

Alumina 5000 1

Perspex 30 1

For most components the size of the plastic zone is fairly small but concerns mustbe raised for the validity of LEFM in the case of structural steels. In practice theASM standard requires that the crack length a, the specimen thickness B, and the

residual specimen width of a test-piece are all greater than

2.5KI

ys

2

.

This means that, in effect, rp < a/8 for LEFM to apply. The plastic zone should beless than 20% of the area dominated by the crack tip stresses (rather than the

remote stresses) which is about 10% of the crack length.

Alternatively we can look at the effect of the plastic zone on the fracture stress

f EGcrit

a ry

or

f EGcrit

a f2a /2ys

2

The plastic zone has the effect of dividing by the factor

1 f2

2ys2

For 4.0ys

f

the error is 4%; for 0.6 the error is 8.5% and for 0.8 the error

reaches 15%. Hence the closer the fracture stress gets to the yield stress the

more ductile the failure and the greater the influence of the plastic zone.


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REAL SHAPE OF PLASTIC ZONE

The plastic zone is not going to be circular since the largest shear stresses occur

at 45 to the crack (equations page 10). The exact shape is tricky to calculateand depends on the yield criterion used. Using the Von Mises criterion for yield :

ys 1

21 2

2 1 3

2 2 3

2 12

and substituting the Mode I principal stresses in polar co-ordinates:

1 KI

2rcos

2

1 sin

2

2 KI

2r

cos

2

1 sin

2

03 for plane stress, and

3 2KI

2rcos

2

for plane strain

we are able to solve for rp and obtain the limits of the plastic zone:

rp 14

KIys

2

1cos3

2sin2

For plane stress

rp 14

KIys

2

12 2 1cos 32sin2

For plane strain

plotting this gives the shapes for the plastic zone. Note the value for plane strainwill be smaller by some (1-2)2 which is 0.16 for = 0.3. Thus the plastic zone

is of a slightly different shape and smaller in size for the constrained central partof the crack.

Diagram of the plastic zone and the effect of through thickness crack.

Plane stress at outside edge

Plane strain in centre


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Plastic Zone shape for Mode I, II and III crack opening, calculated from

von Mises yield criterion.

Similarly the plastic zone size and shape can be derived for the other crack

opening modes and these are shown in the above Figure. In general the mostlikely cause of crack growth is mode I opening, and consideration of this is able tosolve most problems.

Again it must be emphasized that the exact solution depends on the plasticity ofthe material and that there is a gradual transition from plane stress to planestrain. A high work-hardening rate reduces the plastic zone size as more stress

can be sustained by the plastic material. When the plastic zone size becomescomparable with the thickness of the specimen, plain strain is not achieved at thecentre of the crack. However, provided the plastic zone size is small compared to

the thickness the stress intensity factor KIcprovides a reasonable fracturecriterion.

As the thickness decreases the measured KIc increases from a plane strainplateau value to a higher value characteristic of plane stress. Thus to define KIcasmall plastic zone size and plane strain conditions are required. But use can bemade of LEFM in situations of plane stress i.e. thin plates, provided the values ofKIcthat are used are found in material of similar thickness, In these

circumstances KIc is not a material constant as it varies with the dimensions ofthe specimen.


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KIc

Specimen Thickness

Plane strainPlane stress

Diagram showing the effect of specimen thickness on the critical stress intensity.

The constraint at the centre of a thick sample causes the crack to progress the

furthest at the centre of the crack and the sides fail by plastic shear forming twolips which will point up or down randomly as in the cup and cone fracture. Thecentre part of the crack will be normal to the tensile axis on average, (this masksvalleys and ridges on a smaller scale). As the load on the sample increases theplastic zone size increases and the width in plane strain decreases. Eventuallythe plane stress conditions extend across the sample and a diagonal shear failureresults.

This leads to the kind of fracture surface seen below where the crack starts at anotch propagating by ductile cleavage at right angles to the stress but as thestress increases the area of plastic shear failure gradually takes over.

Notch

Shear Failure - 45

Square failure - plane strain


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R AND G CURVES:

The material resistance to crack extension, R, consists of the energy to create

two new surfaces, 2s together with any mechanism which absorbs energy as thecrack grows. In the case of brittle fracture R does not depend on the size of the

crack, but where plastic work is done developing a plastic zone R may well varywith the crack size, increasing or decreasing. The increase could result from an

increase in the plastic zone size as we saw on the previous page. Initially theconstraint due to the thickness of the specimen inhibits plastic flow, restricts thesize of the plastic zone and keeps R low. As a plastic zone develops at the sidesof the sample R increases reducing the area of ductile cleavage until the entirecrack fails by shear. At this point R reaches a maximum value.

[Alternatively, a decrease could result from the strain rate sensitivity of the flowstress reducing the plastic zone size as the crack grows faster.]

G varies with the size of the crack and the geometry of loading. For fixed grips

the load drops as the crack extends and thus the energy release rate, G, willdrop. But for the same specimen at fixed load, G increases as the crack grows.

G

a

LOAD CONTROL

STRAIN CONTROLFIXED GRIPS


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MEASURING G:

Consider two simple situations, a fixed strain where a growing crack reduces the

load (strain control) and a fixed load where the crack growth increases the lengthof the specimen (load control).

G 1

B

dU

da

u

for strain control and

G 1

B

dU

da

P

for load control.

*Note U = potential energy and u = displacement and P = load.

Consider a plate, thickness B, loaded with a force P. This contains a crack lengtha and as a result of the crack the plate has extended a distance u. The crackextends by a. Under load control the specimen lengthens by u, and the work

done by the external force is UF = - Pu. The extra work stored elastically by

virtue of the change in crack length and the consequent change in specimen

length UE = 1/2Pu. Thus half the work done is stored in the regular way as inan un-cracked body and the rest is released as the elastic response of the bodychanges as a result of the crack growth. Under strain control the load is reducedby P and the energy released: UF = -1/2uP as no external work is done (P is

negative).

LC:

dUE 1

2PduPdu

1

2Pdu SC:

dUE 1

2udP

LC:

GBa 1

2Pu SC:

GBa 1

2uP

We now introduce the Compliance: the inverse stiffness C = u/P.

LC:

G P

2B

du

da

P

P

2B

du

dC

P

dC

da

p

P2

2B

dC

da

p

SC:

G u

2B

dP

da

u

u

2B

dP

dC

u

dC

da

u

P2

2B

dC

da

u

The expression for G is the same in both cases.

The compliance depends on the specimen shape, in particular on the crackgeometry and length, remember the sample is assumed to be elastic at all points.

By measuring the compliance as a function of the crack length the energy releaserate can be calculated from the load P.


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Lets look at this graphically: for a specimen under strain control (the grips arefixed) the crack growth causes a fall in the external force P which is equal to theenergy released by the crack in growing a. This is equal to the area of the

shaded triangle OAC.

a

P

P

u

Pdu

du

dUE = 1/2Pdu

a

a+da

Fixed Load

A B

O

C

For Load control, the specimen extends at fixed load and the energy released isthe area of the triangle OAB. Thus the only difference between the two cases isthe area of the triangle ABC which is of the order 1/2Pu and approaches zero in

the limit.

Thus the value of G depends only on the geometry of the sample: shape, cracklength etc, and the loading, P.

-


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MEASURING R:

For brittle materials R does not change as the crack grows and failureoccurs when the stress rises to the point where G equals R.

The R curve can be measured from a plot of load P against extension u, using the

gradient of theunloading line at any point to give the compliance as the crack

extends.

GP2

2B

dC

da

u

For a rising R curve G must exceed R at any crack length, but as the crack growsR can exceed G. Hence, for fast fracture, G must increase with the crack lengthfasterthan the resistance to crack growth. Fast fracture will occur when dG/da >

dR/da. If dG/da = dR/da the crack will continue growing in a controlled manner(so-called stable crack growth).


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MEASURING KIc

In principle k can be measured from the load at failure and the crack length in astandard sized specimen containing a sharp crack grown usually by fatigue.However, for the test to be valid three criteria must be satisfied:

the specimen must be large enough for the plastic zone size to be a smallproportion of the sample and we have the criterion for the dimensions a, B

and W discussed earlier: a, B and W

2.5KI

ys

2

The maximum fatigue stress intensity K is less than 80% of KIc

the crack is still roughly in the middle ofthe sample, 0.45 a/W 0.55.

If the testpiece were entirely elastic and the load displacement curve would be

linear, it is generally not as the tip of the crack begins to yield. The value of theload, PQ, to be used to assess KIC is taken as the point at which the curve crossesa line drawn with a gradient 95% of the initial tangeant. Sometimes there is asmall amount of unstable crack growth prior to failure at a higher load, pop-inbehaviour. In this case or if the sample fails before a 5% deviation from linearity,the pop-in stress or the ultimate stress prior to failure are used.

The provisional value of KIc, KQ can then be calculated from the equation:

KQ PQ

B Wf a / W

where f(a/W) is a dimensionless function of the specimen dimensions specific tothe testpiece design. These are all set out in the ASTM standard E399. As anexample, for the most common compact specimen testpiece the equation is:

f a / W 2a / W1a / W 3/2

0.8664.64 a / W 13.32 a / W 2

14.72 a / W 3

5.6 a / W 4


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ELASTIC PLASTIC FRACTURE MECHANICS

The requirements for the minimum specimen test-piece size for LEFM to be validare very stringent for ductile materials. In fact the size of test-piece needed toproduce a valid and representative value ofKIcare such that large amounts of

material and huge machines are required for testing. More importantly, the scalecould well exceed the size of the component the results are to be applied to.Under these circumstances we still need a measure of the fracture toughness ofthese materials in order to predict and avoid possible failure. Two methods have

been developed which enable small scale testing to be applied to the failure ofductile materials. These are the Crack Opening Displacement and the J Integralmethod.

CRACK TIP OPENING DISPLACEMENT

Back in 1961 Wells had been trying unsuccessfully to obtain reliable KIc

measurements for ductile steels, when he noticed that the crack tips showed

considerable blunting which increased with the toughness of the material. Heproposed measuring the critical diameter of the crack tip and using this directlyas a measure of the toughness. We will see that for limited plastic zone size thecrack tip opening is related directly and simply to the LEFM energy release rate,but the really useful extension of this to a much larger plastic zone size was atthat point purely empirical. It has since been demonstrated rigorously that the

use of the CTOD is valid even for very extensive plasticity and the method is nowwidely used to test and design components.

Additional crack opening as a result of plasticity at crack tip.

We saw earlier that the effect of a plastic zone at the crack tip is to extend theeffective length of the crack by ry~ half the diameter of the plastic zone. Hencethe opening of the crack at its real tip can be approximated from the calculatedelastic displacements of the virtual (extended) crack evaluated at a point some ryfrom the virtual crack tip. See Figure above.


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The CTOD is given by double the displacement uyy in the tensile direction, for

plane stress this is given by the equation:

uyy KI

2

r

2

sin

2

12cos2

2

where

3

1

for plane stress

evaluating this at ry from the crack tip = 180:

uyy KI 1

2

ry

2

and substituting for the plastic zone size from the Irwin value (second estimate,

page 18):

ry 1

2

KI

ys

2

gives:

uyy

1 2

KI

2

2ys

where

1 2

4

1 1 E

4

E

and hence

2uyy 4

E

KI2

ys4

G

yswhere G is the energy release rate.

Again the Dugdale model gives a similar result:

G

mys

where m is a constant 1 for plane stress and 2 for plane strain.

Remember that this is all derived from the elastic solution surrounding a small

plastic zone (page 10) but it has since been demonstrated from plasticity theorythat this is generally true even if the plastic zone is extensive. The critical valueof the CTOD thus gives a reliable measure of the fracture toughness of thematerial. Clearly this will be a function of the specimen thickness but providedthe thickness of the test-piece is similar to the component the test result can be

used.


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MEASURING CTOD

This is very difficult to measure directly and is usually inferred from the width of

the crack opening V of a three point bending specimen. It is assumed that thespecimen behaves as a rigid hinge pivoting about some point in the uncracked

ligament of the specimen the displacement is then proportional to V:

Wa

V

Wa a

where is a dimensionless constant between 0 and 1.

CTOD measured from a three point bend specimen.

Painstaking experiments measuring the value of V and then by sectioning the

crack established this relationship. But beware - it depends on the specimenthickness and the width of the slot and the length of the crack.

There are four values of recognised by the ASTM standards:

i the CTOD at the onset of stable ductile crack growth.

c the CTOD at the onset of unstable cleavage failure,

u the CTOD at the onset of unstable crack growth following extensive

ductile stable crack growth m the CTOD at maximum load where the specimen does not break.

The first is hard to detect; the only clue in the load curve being a slight change ingradient. The next two are identified by the failure of the sample and the last bya maximum in the load curve without the failure of the sample.

W a


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V

LOADP

Vc Vi

Vm

Vi

Vu

cleavage

stable crack

growth +cleavage

stable crackgrowth +plasticcollapse

Mouth Opening Displacement v Load curves.

J INTEGRALS

The J integral is the equivalent of the G for the elastic-plastic case. It is the rateof energy absorbed per unit area as the crack grows; it is not however the energyrelease rate because the plastic energy is not recoverable as it would be in theelastic case. The definition is:

J dU

dA

where U is the potential energy of the system and A the area of the crack.

Energy release rate for non-linear deformation.

An analogy with the Linear elastic case can be made; compare the Figure above

with those on page 25. The stress strain curve is no longer linear, but the areaunder the curve represents the work done in extending the cracked body (withoutextending the crack).


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Plotting two curves for specimens differing only in the length of the crack, a anda+a, the energy required to grow the crack is the difference in the areas under

the two graphs shaded in the Figures on page 25. Since the area decreases asthe crack grows dU/da is negative and J =-dU/da at unit thickness. Although thisis the same as the definition of the energy release rate we used earlier, the J

integral for the plastic case does not represent the energy released as the crackgrows because much of the energy used performs plastic deformation. This is

fine so long as you are just loading the specimen but becomes tricky if you tryand reverse the stress.

The term J integral comes from the property of J which can be expressed andevaluated as a closed line integral around the crack tip. J is the strain energydensity within the line minus the surface integral of the normal traction stress

forces normal to the surface defined and is independent of the path the integraltakes.

y

x

Diagram showing the line integral around the crack tip J integral.

It can be evaluated experimentally by measuring the stress strain curves for a

number of identical specimens containing cracks of different lengths and plottingthe area under the graph U for each specimen as a function of the crack length

and thus evaluating dU/dA and hence J. There are also specific specimengeometries (deeply double notched and nothed three point bending specimens)that allow J to be measured from a single specimen.

These experiments allow J to be plotted as a function of the crack extension.Thus although J is defined in similar terms to the energy release rate G, andindeed reduces to G for linear elastic behavior, J for elastic-plastic materials iscloser to R, the resistance to crack growth, in both interpretation and form. Thecurve plotted against the crack growth from the original crack length a, showsthree distinct regions; an initial zone where the original crack blunts but does notgrow and the curve rises steeply, a secondary region initiating at JIc, where a newcrack nucleates and grows developing the elastic-plastic zone at the crack tip,

until finally steady state crack tip conditions are achieved and the crackpropagates at a constant value of the J resistance JR.


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JR

Crackblunting

FractureInitiation

Steadystatecrack

growtha

A

C

B

A B C

Diagram indicating the J curve during crack growth.

The validity of this approach has limits, just as the LEFM has. These are reached,

in general terms, when the extent of plastic yielding becomes a large proportionof the remaining ligament length. At this point a single parameter for crackgrowth is not sufficient and even more complicated analysis is necessary.


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FRACTURE MORPHOLOGY

DUCTILE FAILURE:

What do we mean by ductile failure? You are familiar with ductile failure in uni-

axial specimen characterised macroscopically by cup and cone failure and on amicroscopic scale by the formation and coalescence of voids generally nucleatedat second phase particles. This occurs after the point of plastic instability hasbeen reached when the rate of work hardening can no longer compensate for the

increase in the stress as the section decreases. Voids nucleate and grow mostrapidly in the centre of the sample where the state of triaxial stress exists. Thesegrow and coalesce to produce a circular internal crack which grows, and finallyfails by shear in the plane stress outer regions of the sample. Where voidformation is difficult, (for example in pure metals) much more ductility isobserved and the sample can thin almost to a point before failure occurs.

Diagram showing cup and cone failure in tensile specimen

Voids almost always nucleate at second-phase particles either by decohesion atthe interface or by fracture of the second phase or inclusion. A number of modelshave been developed which look at the effect of dislocation pile-ups at second-phase precipitates formed during plastic flow as the trigger to void nucleation butfail to predict the observation that voids appear to nucleate most readily at larger

particles. This is not entirely surprising because the largest precipitates are likelyto be those with the highest interface energy and thus the largest incentive toreduce surface to volume ratio, and, in addition, are also those most likely tocrack under extensive plastic flow in the surrounding matrix. This latter process

is the most likely to occur where large precipitates are present and can be readilyobserved.

The 45 sides of the cone fail last as the central crack propagates outwards. In

the absence of general yielding across the full remaining section of the samplethe progress of a crack by ductile means relies upon the nucleation and growth of

voids ahead of the crack tip. The stress ahead of the crack tip is raised to about4 times the stress at approximately two times the crack tip openingdisplacement or CTOD from the tip. Voids form in this area of raised stressahead of the crack tip.


37/90


Once formed, the voids grow, becoming elliptical and undergoing extensiveplastic flow at the sides. The ligaments between the voids fail by shear on theplane of highest shear stress at 45 to the tensile axis.

CLEAVAGE FRACTURE IN DUCTILE MATERIALS.

The cleavage fracture surface is characterised by a planar inter-granular crackwhich changes plane by the formation of discrete steps. Facets correspond to theindividual grains and in single crystals an entire slip plane can consist of onefacet.

Facetted brittle failure showing river lines.

The steps or river lines on the facets converge and eventually disappear in thedirection of crack growth. They are formed at a grain boundary where thecleavage plane in one grain is not parallel to the plane in the adjacent grain; the

difference being accommodated by a series of steps. These gradually diminish asthe crack propagates adopting the cleavage plane of the new grain before beingre-formed at the next grain boundary.


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If a cleavage crack is to propagate across a grain boundary distinct new cracksmust be nucleated ahead of the interface before sufficient plasticity in thematerial is achieved to relieve those stresses.

Conditions favoring brittle fracture are:

high yield stress,

reduced slip systems (HCP and BCC metals, low temperature),

high constraint (plane strain) and rapid deformation.

However, for metals, in particular for iron , it has been shown that the fracturestress follows the value of yield stress measured in compression (even though intension the material demonstrates brittle failure). For small grains sizes yieldingprecedes failure, at larger grains sizes the two occur together.

At the tip, the crack becomes blunted through plasticity and thus the potentiallyvery high stresses are reduced (see next section). As a result the stressesachieved ahead of the crack tip do not in effect exceed 3-4 times the yield stress.This is way below the theoretical strength of most materials:

Ec

Hence the crack cannot simply propagate as it would in a brittle ceramic. (e.g.the wedging discussed on page 5. There must be a crack or defect ahead of thecrack to further raise the stress and propagate the crack if cleavage is to occur.Under conditions of plane strain i.e. constraint, the critical length for a crack fromthe Griffiths criterion is:

acrit

2Es

1 2 f2 0.3m

where, for example in iron, f= 1GNm-2 and E = 200GNm-2, and s = 2Jm

-2.

Hence some plasticity at the crack tip is necessary to form cracks of roughly thissize in order to propagate the crack further. A number of mechanisms by whichmicro-cracks can form have been proposed and are illustrated on the next page.

The micro-crack is limited to a single grain due to the difficulty in propagatingacross the boundary. Hence the stress intensity ahead of a micro-crack is limitedby the (grain size), this limits the stress to nucleate further cracks and

propagate the failure. This results in a Hall-Petch type relationship between thefailure stress and the grain size:

f Egb

1 2 d

1

2

where gb is the plastic work to propagate across the grain boundary and

generally exceeds the usual p term.

There are other mechanisms by which grain refinement to affect the fracture

stress; in mild steels the cleavage fracture is controlled by the fracture of grain

boundary carbides, and an increase in the overall grain boundary area withsmaller grain size leads to smaller carbides and thus a higher fracture stress.Grain size is hence the one of the best strengthening mechanisms as it increasesboth strength and ductility.


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BRITTLE DUCTILE TRANSITION.

Macroscale:

The brittle ductile transition represents the change from general plastic yieldingto the propagation of a distinct crack this so-called brittle failure can be very

ductile and the fracture surface show evidence of extensive plasticity.

The brittle ductile transition is governed by the macroscopic yield in thespecimen, not what is going on at the crack tip. Hence values depend, withinlimits, on the particular geometry of the specimens. Tests such as the impact

test of which there are several standards (Charpy, Izod etc) provide relativerather than quantitative data. They are nevertheless extremely useful as theyare quick and simple to perform can be compared with reference data to provideexcellent quality control.

If the energy absorbed by rapid failure is plotted against the temperature forsteels a transition is observed from a high to a low value over a limited

temperature range.

Temperature

Energy

absorbed

% Cleavage failure

Energy absorbed

FATTNDT

Two of the transition temperature defined are: the nil ductility temperature where

the curve just begins to rise, and the fracture-surface appearance transitiontemperature, FATT, based on 50% of the surface being cleavage failure. Theformer corresponds to the point at which general yield occurs throughout theremaining width of the sample.

Factors promoting cleavage failure are:

1 high yield stress large amount of stored elastic energy

2 large grain size large build up of stress from pile-ups

3 coarse carbides can crack

4 deep notches - constraint

5 thick specimens (plane strain).


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At the nano-micro scale:

Fracture of very small components is crucial to the development of small devices

and it is here that much interest in fracture is currently focused. Here plasticity isalso crucial, particularly in materials with limited dislocation mobility (Si, Ge, Fe,

Cr, Al2O3, and inter-metallics) essentially everything other than fcc metals. Allthese materials display very brittle behaviour at low temperatures and atransition to a more ductile behaviour as temperature rises.

Rice introduced the concept that brittleness was determined by the competition atthe crack tip between the generation of dislocations in the very high stress fieldat the crack tip and cleavage. His paper of 1974 explains the issue very lucidly(skip the mathematics in the middle)J.R. Rice and R Thomson, Phil Mag 29, 1,

p73, (1974), with a more modern interpretation given by J.R. Rice, Journal of the

Mechanics and Physics of Solids, V.40, Iss.2 p.239-271 (1992).

This is demonstrated by a series of experiments performed by Prof Steve Robertson pure iron single crystals. (Acta. Mat. 56 (2008) 5123)

4Pt bending with pre-cracked single crystals of specific orientation (2 slipplanes at 45 to the crack tip)

Strain rate varied from 4 x 10-3 to 4 x 10-5 s-1

KIc calculated from failure stress and geometrical factors

DBT indentified from examination of the fracture surface and evidence ofslip bands

Plotting 1/TDBT against strain rate shows an Arhenius relationship

Activation energy correlates very well with that for dislocation movement

The DBT decreases from 130K at the lowest strain rate to 154K at the highest.

The observed behaviour can be modeled very accurately by dislocationdynamics. This means calculating the distribution and movement of dislocationsduring the test from their initial positions, the complete stress field and anexponential equation for dislocation velocity.

Essentially the DBT occurs when the shielding effect of the dislocations on thetwo slip planes (i.e. the elastic stress fields from those generated) reduces thestress at the crack tip sufficiently rapidly to prevent the stress at the tip reachingthe cleavage stress.


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FATIGUE

Fatigue is damage (usually failure) caused by oscillating stress below the fracturestress. 90% of all mechanical failures can be attributed to fatigue. Paradoxically,although the stress is below the yield stress, fatigue is essentially concerned with

the generation of defects by plastic flow and the movement of dislocations.

Time

Stress

max

m

min

0

m

a

The diagram above defines some of the variables used to describe a fatigue test

run under stress control: the stress range a, mean stress

m. the R ratio R = min/max .

Similar definitions apply to tests where the strain on the sample is controlled andthe maximum stress may vary through the test.

Real fatigue situations cover a baffling range of variables; examples include highfrequency mechanical fatigue for example in a crankshaft, to low frequency

pounding of a north-sea oil rig structure in a highly corrosive environment, tothermal fatigue caused by the periodic heating and cooling in the turbine of atransatlantic jet engine. We need to understand fatigue so that we are able to:

i) predict the engineering life of these components,ii) design structures and materials which maximise economic life.

Factors affecting fatigue which we will consider in varying degrees of detail are:

Mean stress m

Stress amplitude

Frequency

Waveform Temperature Temperature variation

Environment corrosion and oxidation

Surface finish Coatings Microstructure


43/90


Test procedures have been developed which address these variables and by theuse of a number of mostly empirical laws these are able to provide some degree

of predictability in most situations. Fatigue conditions fall into a number ofregimes:

High Cycle Fatigue HCF: Low amplitude stresses induce primarily elastic strains

which results in long life, i.e. endurance in excess of 10,000cycles

Low Cycle Fatigue LCF: Considerable plastic deformation during cyclic loadingresults in an endurance limit below 10,000 cycles and behavior dominated by

plastic deformation.

Thermo-mechanical Fatigue TMF: varying both stress and temperature to

give strain cycles in phase, out of phase (and all things in between) with thetemperature cycle.

APPROACHES TO FATIGUE

We can break Fatigue in ductile materials into several stages:

1. Initial micro-structural changes leading to the nucleation of permanentdamage

2. Nucleation of the first micro-cracks

3. Growth and coalescence of these flaws to produce a dominant crack.

4. Stable propagation of the dominant crack.

5. Failure

Macroscopically there are ambiguities in defining the initiation and growth stagesof cracks depending on the resolution of the techniques being used toinvestigate. Generally stages 1-3 constitute crack initiation and stages 4-5 crackgrowth.

Depending on the conditions, these stages occupy widely differing fractions of the

sample life and thus require different strategies to determine life. The methodadopted also depends on the consequences of failure.


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TOTAL-LIFE OR SAFE-LIFE:

This strategy is to predict the total life and retire the component at a fixedproportion of this, to include a considerable margin for error. The aim is to retirethe component before a crack forms and it is used where fatigue failure would

result in component failure. Total-life can be wasteful as much useful life remainsunused where the scatter in the data is large.

This approach focuses on predicting the number of cycles to failure, N for an

initially un-cracked specimen. This is most appropriate where the initiation of thedominant crack occupies the majority of the total life (as much as 90%). For HCFwhere the stress range is low and the stresses principally elastic, the stress rangeis used to characterise the component and produce a reference S-N curve. For

higher stresses resulting in LCF plastic strain is extensive and the strain range istypically (but not always) used.

DAMAGETOLERANT OR FAIL-SAFE:

This approach recognises that all structures contain defects and that these growat stable and predictable rates. The strategy involves periodic inspection of thestructure and repairs or replaces components as cracks are found. This isgenerally used where failure would not result in component failure due tostructural redundancy. A greater proportion of the useful life is used and the riskof wrong assumptions in the predictive process are dimished.

Thus if the maximum size of the initial defects in the structure is known (amax) the

interval between inspections is determined by the time predicted for this crack toachieve critical size (t1), (we will quantify this later) The component may surviveseveral iterations (two in the case below) before being replaced.

LEAK BEFORE BREAK:

A special case of the fail-safe approach widely used for pressure vessels andpipes. The thickness and properties of the vessel are arranged so that a through-thickness crack does not propagate catastrophically. This means that the crackwill be below the critical size for the stress on the vessel. Such a leak can be

detected and repaired without the severe consequences of the rupture of thevessel.


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TOTAL LIFE APPROACH

If we perform a series of tests at varying stress ranges and plot the number ofcycles to failure the life increases as the stress range decreases. Some materials(typically low alloy steels and Titanium alloys) show an asymptote to a fatigue

limit, otherwise (high alloy steels and aluminium), an endurance limit is set.

ln N

Fatigue limit

10 Endurance limit7

BASQUINS LAW

The curve can be approximated by an empirical expression due to Basquin:

bffa N22

Nf is the number of complete cycles to failure.

where f is the fatigue strength coefficient f the static fracture strength and btakes the value 0.05 to 0.12 for metals.

COFFIN MANSON LAW.

Under conditions of high plastic deformation we have low cycle fatigue conditionsand for strain controlled tests, Coffin and Manson independently noted anempirical relation very similar to Basquins law.

The total strain amplitude can be split into plastic and elastic components:

222

pe

where the plastic component is linear when plotted against the log (number ofload reversals), 2Nf:

cffp N22

Here f is the fatigue ductility component and roughly equal to the failure ductility

in tension, and c takes the value 0.5 to 0.7 for metals.

Adding in the Basquins law for the elastic (high cycle fatigue) component wehave:

c

ff

b

ff

N2N2E2


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Plotting log() against log (2Nf) gives two distinct regimes, at low strain and

long life the gradient b (-0.1) dominates, HCF conditions, and at high strain andshort life the gradient is c (-0.5). The transition is gradual but extrapolating the

asymptotes allows a transition number of cycles, 2Nt, to be identified.

log 2Nf

log c

b

f

2Nt

Note: fatigue is inherently variable variation in life of 100% is not unusual for

nominally the same test. This is masked by the widespread use of log plots.

The intercepts of the two parts of the curve correspond roughly to:

1. LCF: the total strain, plastic and elastic, at failure.2. HCF: the elastic component of the strain at failure

Lets put some figures in here:

E f f b c

Aluminium 7075 72GPa 193MPa 1.8 -0.106 -0.690

Steel 0.15%C 210GPa 827MPa 0.95 -0.110 -0.640

Aluminium: 69.0f106.0

f N28.1N272000

193

2

(HCF intercept 666 times less than the LCF intercept - note log scale)

Steel: 64.0f11.0

f N295.0N2210000

827

2

(HCF intercept 240 times less than LCF)


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TOTAL LIFE APPROACH - COPING WITH FATIGUE VARIABLES

There is a huge number of variables in fatigue far to many to construct S/Ncurves for all combinations even if they did not change during the lifetime of thecomponent. The challenge is to understand how the damage produced by fatigue

varies with these parameters and adds together over a complex life cycle.

The effect of increasing the mean stress is to decrease the fatigue life. Severalrelations exist to link the stress range and the mean stress for a given life. The

simplest are linear extrapolations indicating that the sample will fail at the staticyield stress in the absence of a stress range and at the fatigue strain at zeromean strain.

a a | m0 1my

Soderberg: original and most conservative

a a | m0 1mTS

Goodman relation: good for Brittle materials

conservative for metals

(Other expressions exist giving non-linear extrapolations


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GOODMAN DIAGRAM

The effect of mean stress and R value can be expressed on a Goodman diagramshown below:

TEMPERATURE

It is possible to adjust for temperature where the nature of Fatigue does notchange by normalizing with the yield stress. By plotting /y fatigue curves from

different temperatures can become very similar.


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COPING WITH VARIABLE STRESS - MINERS LAW:

In real situations components very rarely experience constant regular damage.The level of stress or strain can vary throughout life and the simplest way ofdealing with this is by the use of Miners law. This proposes that the life of acomponent experiencing fatigue at various stress amplitudes can be assessed by

expressing the number of cycles at each amplitude as a proportion of total lifeand summing the fractions. When the fraction reaches 1, the fatigue life isexhausted. The order of exposure is not taken into account.

time

m

1i fi

i 1N

n Miners Law

This is useful as a first approximation but has serious shortcomings. The most

important being that no account can be taken of the impact of prior damage onthe later exposure at a different stress (or strain) amplitude. In particular thebalance between crack initiation and crack growth can vary considerably withstress, thus brief exposure to high amplitude may nucleate damage which at a

lower stress would not occur until a much later stage and thus accelerate thedamage rate at a subsequent lower stress. Conversely, early exposure to lowstress amplitude may strain harden the material and thus prolong life during laterhigh amplitude exposure. This emphasizes the importance of looking at thespecific mechanisms of damage and how it accumulates in the material.


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DEVELPOMENT OF MICROSTRUCTURE DURING FATIGUE

For some situations the loading conditions are controlled by the amplitude of thestrain rather than the stress. This is reflected in the tests which are done understrain control. These are also most likely to be the conditions where plasticdeformation forms a considerable proportion of the strain, LCF.

Where the strain is kept constant the stress can either increase (cyclic hardening)

decrease (cyclic softening) or stay the same.

Typically materials harden if 4.1ys

UTS

and soften if 2.1

ys

UTS

To understand why this occurs we need to consider dislocation microstructure ofthe material.

From the above materials where the initial state is highly work hardenedthe dislocation density is high, the effect of the cyclic strain is to allow the

rearrangement of the dislocations into stable networks, reducing the stressat which the plastic component occurs, and thus the effective stress.

Conversely where the initial dislocation density is low, (soft material) the

cyclic strain increases the dislocation density increasing the amount ofelastic strain and the stress on the material.

For a given alloy both hard and soft materials tend to a stable dislocationconfiguration. For example, detailed work on the development of dislocationconfigurations in copper and shows that a stable Laberynth structure develops(see the figure on page 52 from Suresh, chapter 2)

This results in a hysteresis loop which becomes stable at some point during thetest. When this stable loop is plotted as a function of increasing stress (orstrain) the locus of the maximum values from a series of tests defines a cyclic

stress strain curve.


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BAUSCHINGER EFFECT

Strain control first cycle showing the Bauschinger effect.

During cyclic deformation the material can retain a memory of the initial plasticstrain which reduces the stress at which plastic yield occurs in the reverse cycle.

This effect can persist for many cycles and is known as the BAUSCHINGEREFFECT. This reversible but plastic deformation can occur by dislocation pile-upsat, for example, incoherent or semi-coherent precipitates exerting a back-stresswhich assists plastic yield in compression. This reduces the yield stress incompression.


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The picture above shows a single crystal superalloy CMSX-4 fatigued in LCF at750C and interrupted at about half the expected life. In this section cut on theprimary slip plane dislocation loop enter the phase precipitates trailing Anti-

Phase Boundary faults under max stress the loops expand contracting as the

stress decreases.


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THE EFFECT OF STACKING FAULT ENERGY:

The material response is closely linked to the stacking fault energy since thisgoverns the ability of the dislocation to cross-slip between planes and thus formstable cell structures.

High SFE easy cross-slip rapid formation of stable cell structure.

For high SFE materials the cell size is a decreasing function of the strain rangeand ultimately does not depend on the starting microstructure. Very low SFE

e.g. Cu 7.5% Al, materials do not form stable cell structures, the highlydissociated dislocations being arranged in planar arrays where the spacingdepends on the initial state. The microstructures and lives are thus very sensitiveto the prior deformation state.


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CYCLIC SOFTENING OF PRECIPITATION HARDENED ALLOYS:

Another cause of cyclic softening of major importance is the cutting of coherentprecipitates in precipitation-hardened alloys. Small coherent precipitates providevery effective hardening in aluminium alloys and in nickel based superalloys. The

small size maximises the cutting/bowing stress for dislocations in the matrix andthe coherency enhances the stability of these small precipitates. However when adislocation does cut the precipitate the fault produced in the precipitatesdecreases the stress for the following dislocation since the energy penalty of the

fault no longer applies and indeed may be negative. Thus slip is concentrated innarrow slip bands cutting the precipitates in two. These smaller precipitates maydissolve in these areas leaving the un-strengthened matrix vulnerable to highplastic deformation and early crack formation.

Here cutting of precipitates early in this test (TMF of Nimonic 90) has caused the

to dissolve leaving precipitate free channels in the alloy. The precipitates arevisible from the dislocations wrapped around them

(a) OP (0.4)


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STRUCTURAL FEATURES OF FATIGUE CRACK INITIATION:

Here we look briefly at the effect of surface condition, the evolution of damage,and the effect of coatings and surface treatments on the initiation andpropagation of fatigue cracks.

INITIATION BY DEFORMATION

Fatigue failures can occur at stress of 1/3 the tensile yield stress, yet the

nucleation of cracks requires that there be local yielding. We thus needheterogeneous nucleation sites for cracks within the structure.

Pre-existing defects such as inclusions, porosity, surface damage

Defects generated during cyclic straining for example at stressconcentrations: persistent slip bands, Fracture of carbides, oxidation ofcarbides.

In pure metals the major source of fatigue cracks is the persistent slip bands or

PSBs: so called because traces of the bands persist even after surface damage ispolished away. The plastic strain in the PSB is 100 times greater than that in the

surrounding material and results from specific arrangements of dislocations asparallel walls with relatively low dislocation density between. An equilibrium ismaintained between nucleation and annihilation of mobile edge dislocations

bowing out from the walls. Thus the cyclic strain is concentrated in these zonesleading to their persistence. Although the strain is reversed within the PSB thedistribution is not even and this leads to the formation of intrusions andextrusions where PSBs intersect the surface. These may act as nucleation sites

for cracks. The initial stages of crack growth therefore often follow the slip planesand lie at 45 to the tensile axis (see later).

Diagram showing the formation of intrusions and extrusions at a persistent slip band.

In the vast majority of cases fatigue initiates on the surface, however cracks cansometimes initiate internally at defects cracked precipitates of internal porosity.The crack then grows under vacuum until it reaches an external surface. This

gives a characteristic circular area on the fracture surface. Once the crackbecomes a surface crack and air is admitted and the stress intensity increasesand the growth rate increases. This sometimes leads to immediate failure.


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EFFECTS OF OXIDATION AND CORROSION

The absence of a fatigue limit is normally and indication of material which isimmune from corrosion or oxidation effects otherwise the mere passage of timewill eventually allow the initiation and propagation of cracks even at very low

stress.

At high temperatures oxidation at grain boundaries, Carbides or as in the singlecrystal above, in areas between the dendrites where the composition varies


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slightly, results in crack initiation. The brittle oxide layer cracks and raises thestress at the tip sufficiently for the crack to propagate into the substrate thus

allowing further oxidation.

Coatings with different mechanical properties to the substrate can acceleratecracking by promoting rapid initiation.

CoNiCrAlY coating on IN738 Tested inTMF 300C 850C

DESIGN AGAINST FAILURE

Where initiation occupies most of the life of the sample, initiation control, any

measure which reduces crack nucleation will extend life.

Surface damage, even scratches can act as stress concentrators and lead to localplastic deformation and crack initiation. High cycle fatigue is very sensitive to

surface finish and can be extended by polished surface finish.

Corrosion protection to suppress the formation of cracks at interfaces by

preferential attack

Treatments which induce a residual compressive stress in the surface layer willextend life by reducing the mean stress at the surface and delaying the onset of

cracking. Carburising, Nitriding, shot peening.

Coatings have different stress and/or thermal response to the imposed stress

may crack. The crack can act as a stress concentrator and promote earlycracking. Thermal barrier coatings accelerate HCF failure.

Pores act as stress concentrators and a major source of fatigue cracks in single

crystal superalloys: remove by HIPING.

As with fracture plastic deformation in the matrix can cause cracking of carbidesand the nucleation of cracks.

Crack interface bridging


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DAMAGE TOLERANT APPROACH

The damage tolerant approach recognises that crack initiation may occur early inthe life of the sample or that there are pre-existing cracks which grow in a stablemanner through the life of the component.

Following the development of fracture mechanics for monotonic deformation Parisrecognised in the 1960s that the same concepts could be applied to fatigue toestimate the fatigue crack growth rate and thus predict the time taken for thecrack to reach an unstable size.

If the rate of crack growth is measured and plotted against the K on a log-log

plot the curve takes the general sigmoidal form shown below.

I: Crack initiation,

crack at 45

following slip planes

II: Crack propagatesat 90 to tensile

axis, striationsformed

III: Final rupture

There are threedistinct regions, an initial stage usually showing a threshold value for K, a 2nd

stage where the crack growth rate shows a power law dependence on K only;

and a final stage where the crack growth rate approaches infinity as the K

reaches KIc. The central region is the most useful as it allows the CGR for the

major part of the life to be predicted from a knowledge of the conditions at thecrack tip. This equation is known as the Paris Equation.

mKCdN

da

where m 4 but can vary from 2-7 for various materials.

This implies that da/dN does not depend on the value of R. This is not strictly the

case particularly for low values of R where the crack closes during the cycle (seep59).

Note: Minors Law follows directly from the Paris Law see question sheet 2

log(da/dN)

log K

Fracture

I II III

m

KIc


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CRACK MORPHOLOGY

The three regions can be identified from the morphology of the fracture surface.

I. Initiation; crack initiates at intrusions and follows slip plane at

approximately 45 to principle stress direction. When the length is

sufficient for the stress field at the tip to become dominant the overallcrack plane becomes perpendicular to the principle stress and thecrack enters stage II.

II. Growth typically showing striations for each cycle and beach marks atpoints where conditions changed. Striations may be obscured byclosure damage or by oxide formation at high temperatures.

III. Final failure ductile or brittle rupture associated with fast fracture.

Intrusions andextrusions

Crack -Stage I

CrackStage II

Crack stage I growth at 45 to stress axis, following persistent slip bands,

during stage II turns to growth normal to stress axis.

Formation of striations by ductile flo

Documents

Griffith's Analysis + Fatigue + Fracture