Chapter 9d Fracture

8/19/2019 Chapter 9d Fracture

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FRACTURE

Brittle Fracture: criteria for fracture.

Ductile fracture. Ductile to Brittle transition.

Fracture MechanicsT.L. Anderson

CRC Press, Boca Raton, USA (1995). Fracture Mechanics

C.T. Sun & Z.-H. JinAcademic Press, Oxford (2012).

MATERIALS SCIENCE

&

ENGINEERING

Anandh Subramaniam & Kantesh Balani

Materials Science and Engineering (MSE)

Indian Institute of Technology, Kanpur- 208016

Email: [email protected], URL: home.iitk.ac.in/~anandh

AN INTRODUCTORY E

-

BOOK

Part of

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A Learner’s Guide

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Theoretical fracture strength and cracks

Let us consider a perfect crystalline material loaded in tension. Failure by fracture can occurif bonds are broken and fresh surfaces are created.

If two atomic planes are to be separated the force required initially increases to a maximum

(Fmax) and then decreases. The maximum stress corresponding to Fmax is the theoretical strength t . This stress is given by:

A p

p l i e d F o r c e ( F ) →

r →a0

Cohesive force

0

a

E t TFS

E → Young’s modulus of the crystal → Surface energy a0 → Equilibrium distance between

atomic centres

Fmax

0

This implies the theoretical fracture strength is in the range ofE/10 to E/6*.

The strength of real materials is of the order of E/100 to E/1000 (i.e. much lower in

magnitude). Tiny cracks are responsible for this (other weak regions in the crystal could also be responsible for this).

*For Al:

E=70.5 GPa, a0=2.86 Å, (111)= 0.704 N/m. t = 13.16 GPa

Cracks play the same role in fracture (of weakening)

as dislocations play for plastic deformation.

By Energy consideration

2

E TFS

By atomistic approach

For many metals ~ 0.01Ea0


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Fracture is related to propagation of cracks, leading to the failure of thematerial/component.

If there are no pre-existing cracks, then a crack needs to nucleate before propagation (to

failure). Crack nucleation$ typically requires higher stress levels than crack propagation. A crack is typically a ‘sharp*’ void in a material, which acts like a stress concentrator or

amplifier . Hence, crack is a amplifier of a ‘far field’ mean stress. (Cracks themselves do not produce stresses!). [A crack is a stress amplifier !].

Cracks in general may have several geometries. Even a circular hole can be considered as a

very ‘blunt’ crack. A crack may lie fully enclosed by the material or may have ‘crack faces’connected to the outer surface. Cracks connected with outer surface may be profoundly influenced by the environment.

Crack propagation leads to the creation of new surface area, which further leads to theincrease in the surface energy of the solid. However, in fracture the surface energy involved(the fracture sur face energy ) is typically greater than the intrinsic surface energy as fractureinvolves ‘sub-surface’ atoms to some extent. Additionally, the fracture surface energy may

involve terms arising out of energy dissipation due to micro-cracking, phase transformationand plastic deformation.

Fracture

2a

A crack in a material

Fracture surface energy (f) > Intrinsic surface energy ()

$ Regions of stress concentrations (arising from various sources) ‘help’ in the process.

* More about this soonerClick here What is meant by failure?

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Fracture mechanics is the subject of study, wherein the a materials resistance to fracture is

characterized. In other words the ‘tolerance’ of a material to crack propagation is analyzed*.

Crack propagation can be steady (i.e. slowly increasing crack length with time or load) or

can be catastrophic (unsteady crack propagation, leading to sudden failure of the material)$.

‘What dislocation is to slip, crack is to fracture’.

Under tensile loading if the stress exceeds the yield strength the material, the material

begins to plastically deform. The area under the stress-strain curve is designated as the

toughness in uniaxial tension. Toughness relates to the energy absorbed to fracture.

Similarly, in the presence of cracks we arrive at a material parameter , which characterizes

the toughness of the material in the presence of cracks→ the fracture toughness. In most materials, even if the material is macroscopically brittle (i.e. shows very little

plastic deformation in a uniaxial tension test), there might be some ductility at the

microscopic level. This implies that in most materials the crack tip is not ‘infinitely’ sharp,

but is blunted a little. This further avoids the stress singularity at the crack tip as we shall

see later.

Sharp Crack (tip)Crack after crack tip

blunting process

$ One of the important goals of material/component design is to avoid

catastrophic failure. If crack propagation is steady, then we can

practice preventive maintenance (i.e. replace the component after

certain hours of service) → this cannot be done in the case of

catastrophic failure.

* Amongst its many other goals!


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Breakingof

Liberty ShipsCold waters

Welding instead of riveting

High sulphur in steel

Residual stress

Continuity of the structure

Microcracks

The subject of Fracture mechanics has its origins in the failure of WWII Liberty ships. In

one of the cases the ship virtually broke into two with a loud sound, when it was in the

harbour i.e. not in ‘fighting mode’.

This was caused by lack of fracture toughness at the weld joint, resulting in the propagation

of ‘brittle cracks’ (i.e. crack propagation will little plastic deformation). The full list of factors contributing tothis failure is in the figure below.

It is seen that welding was done for faster production, but this resulted in micro-cracks and

residual stresses, which led to brittle crack propagation. The problem became ‘global’ as

this provided continuity of crack path across plates (so instead of one plate breaking the

entire ship ‘broke’). High sulphur in steel contributed to the brittleness of the plates. Due to the cold sea waters the ships were harboured in, the hull material underwent a

phenomenon known as ‘ductile to brittle transition (DBT)’ (about which we will learn more in this chapter).

Ironically, this ‘death’ of ships lead to the ‘birth’ of fracture mechanics as a systematic field

of study.


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2a

A crack in a material

What is a crack?Funda Check

As we have seen crack is an amplifier of ‘far -field’ mean stress. The sharper the crack -tip,the higher will be the stresses at the crack-tip. It is a region where atoms are ‘debonded’ andan internal surface exists (this internal surface may be connected to the external surface).

Cracks can be sharp in brittle materials, while in ductile materials plastic deformation at thecrack-tip blunts the crack (leading to a lowered stress at the crack tip and further alterationof nature of the stress distribution).

Even void or a through hole in the material can be considered a crack. Though often a crack is considered to be a discontinuity in the material with a ‘sharp’ feature (i.e. the stress amplification factor is large).

A second phase (usually hard brittle phase) in a lens/needle like geometry can lead to stress

amplification and hence be considered a crack. Further, (in some cases) debonding at theinterface between the second phase and matrix can lead to the formation of an interfacecracks.

As the crack propagates fresh (internal) surface area is created. The fracture surface energyrequired for this comes from the strain energy stored in the material (which could furthercome from the work done by externally applied loads). In ductile materials energy is also

expended for plastic deformation at the crack tip. A crack reduces the stiffness of the structure (though this may often be ignored).

Hard second phase in

the material Though often in figures the crack is shown to have a large lateralextent, it is usually assumed that the crack does not lead to an

appreciable decrease in the load bearing area [i.e. crack is a local

stress amplifier, rather than a ‘global’ weakener by decreasingthe load bearing area].


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~

2a a

Characterization of Cracks

Cracks can be characterized looking into the following aspects.

Its connection with the external free surface: (i) completely internal, (ii) internal cracks withconnections to the outer surfaces, (iii) Surface cracks.

Cracks with some contact with external surfaces are exposed to outer media and hencemay be prone to oxidation and corrosion (cracking).We will learn about stress corrosion cracking later.

Crack length (the deleterious effect of a crack further depends on the type of crack (i, ii oriii as above).

Crack tip radius (the sharper the crack, the more deleterious it is). Crack tip radius isdependent of the type of loading and the ductility of the material.

Crack orientation with respect to geometry and loading. We will see modes of loading inthis context soon.


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Mode I

Mode III

Modes of Deformation /

fracture of a cracked body

Mode II

Three ideal cases of loading of a cracked body can be considered, which are called themodes of deformation:

Mode I: Opening mode Mode II: Sliding mode

Mode III: Tearing mode

In the general case (for a crack in an arbitrarily shaped body, under an arbitrary loading), themode is not pure (i.e. is mixed mode). The essential aspects of fracture can be understood by considering mode I.

Modes of deformation of a cracked body (modes of fracture)

Important note: the loading specified and the geometry of the specimen illustrated for Mode II & III above do not give riseto pure Mode II and II deformation (other constraints or body shapes are required).

How many ways are there to load a cracked body?


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Fracture

Brittle

Ductile

One of the goals of fracture mechanics is to derive a material property (the fracturetoughness), which can characterize the mechanical behaviour of a material with flaws(cracks) in it.

Fracture can broadly be classified into Brittle and Ductile fracture. This is usually doneusing the macroscopic ductility observed and usually not taking into account the microscale plasticity, which could be significant. A ductile material is one, which yields before fracture.

Further, one would like to avoid brittle fracture, wherein crack propagation leading tofailure occurs with very little absorption of energy (in brittle fracture the crack may grow

unstably, without much predictability). Three factors have a profound influence on the nature of fracture:

(i) temperature, (ii) strain rate, (iii) the state of stress.

Materials which behave in a brittle fashion at low temperature may become ductile at hightemperatures. When strain rate is increased (by a few orders of magnitude) a ductilematerial may start to behave in a brittle fashion.

Fracture: Important Points

Ductile material : y < f

Promoted by High Strain rate

Triaxial state of State of stress

Low Temperature

Factors affecting(the nature of) fracture

Strain rate

State of stress

Temperature


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Why do high strain rate, low temperature and triaxial state of stress promote brittle fracture?

Funda Check

High strain rate (by not giving sufficient time) and low temperature essentially have a

similar effect of not allowing thermally activated motion of dislocations (i.e. ‘not helping’

plastic deformation by slip). In specific cases some of the slip systems being active at high temperatures may become

inactive at low temperatures.

By triaxial state of stress (SoS) we mean tensile stresses of same sign along ‘y’ and ‘z’ also.

Triaxial SoS does not promote crack propagation, but suppresses plastic deformation (click

on link below to know more). Since plastic deformation is suppressed the crack tip remainssharp, thus promoting brittle fracture.

So for plastic deformation the following order is better: tri-axial


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Considerable amount of information can be gathered regarding the origin and nature offracture by studying the fracture surface. In fatigue failure for instance, we can know the place of origin of cracks, stable crack propagation regime, etc.

The fracture surface has to be maintained in pristine manner (i.e. oxidation, contact damage,etc. should be avoided) to get meaningful information from fractography.

It should be noted that a sample which shows very little macroscopic ductility, may displaymicroscopic ductility (as can be seen in a fractograph).

Truly brittle samples show faceted cleavage planes, while ductile fracture surface displays a

dimpled appearance.

Fractography

Fracture surface as seen in an SEM*

* The Scanning Electron Microscope (SEM) with a large depth of field is an ideal tool to do fractography.


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Behaviour described Terms Used

Crystallographic mode Shear (ductile) Cleavage (brittle)

Appearance of Fracture surface Fibrous Granular / bright

Strain to fracture Ductile Brittle

Path Transgranular (crack propagates through the grains) Intergranular (crack propagatesthrough the grain boundaries)

Fracture can be classified based on:(i) Crystallographic mode,(ii) Appearance of Fracture surface,

(iii) Strain to fracture,(iv) Crack Path, etc. (As in the table below).

Presence of chemical species at the crack tip can lead to reduced fracture stress andenhanced crack propagation.

Presence of brittle phase along the grain boundaries (Fe3C along GB in steel, glassy phase at

GB in Si3 N4 ceramics) can lead to inter-granular crack propagation. This preferred ‘weak’ path along grain boundaries implies low energy expenditure during fracture (i.e. lowfracture toughness).

Classification of Fracture (based on various features)

Brittle cementite along grain boundaries along which

crack can propagate

0.66 nm

1.4 nm

IGF

Grain-1

Grain-2

0.66 nm

1.4 nm

IGF

Grain-1

Grain-2


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Brittle Shear Rupture Ductile fracture

Little or no deformationShear fracture of ductile

single crystalsCompletely ductile

fracture of polycrystals

Ductile fracture of usual polycrystals

Observed in single

crystals and polycrystals

Not observed in

polycrystals

Very ductile metals likegold and lead neck down

to a point and fail

Cup and cone fracture

Have been observed inBCC and HCP metals but

not in FCC metals

Here technically there is

no fracture (there is not

enough material left to

support the load)

Cracks may nucleate atsecond phase particles(void formation at the

matrix-particle interface)

S l i p P

l a n e

Cleavage plane

Types of failure in an uniaxial tension test


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‘Early Days’ of the Study of Fracture

C.E. Inglis A.A. Griffith Stress based criterion for crack growth

(local)

→ C.E. Inglis (seminal paper in 1913)[1]

Energy based criterion for crack growth (global)

→ A.A. Griffith (seminal paper in 1920)[2]

(Work done on glass very brittle material).

[1] C.E. Inglis, Stresses in a plate due to the presence of cracks and sharp corners, Trans. Inst. Naval Architechts 55 (1913) 219-230.[2] A.A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans. R. Soc. Lond. A221 (1920) 163-198. → Fat paper!


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Crack growth and failure

Crack growth criteriaStress based

Energy based Global

Local

Griffith

Inglis

Initially we try to understand crack propagation$ in brittle materials (wherein the cracks are

sharp and there is very little crack-tip plasticity). The is the domain of Linear Elastic

Fracture Mechanics (LEFM).

For crack to propagate the necessary global criterion (due to Griffith) and the sufficientlocal criterion (due to Inglis) have to be satisfied (as in figure below).

The kind of loading/stresses also matters. Tensile stresses* tend to open up cracks, while

compressive stresses tend to close cracks.

Global vs. Local

For crack growth to occur Sufficient stress concentration shouldexist at crack tip to break bonds

It should be energetically favorable

$ Note: the crack propagation we will study in this chapter will be quasi-static (i.e. elastic wave propagation due to crack growth is ignored)* More on this later.

Brittle Materials

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Stress based criterion for crack propagation (Inglis criterion)

In 1913 Inglis observed that the stress concentration around a hole (or a ‘notch’) depended

on the radius of curvature of the notch. I.e. the far field stress (0) is amplified near the hole.[(max / 0) is the stress concentration factor ()].

A ‘flattened’ (elliptical) hole can be thought of as a crack.

cσ σ 210max

0 → applied “far field” stress

max → stress at hole/crack tip

→ hole/crack tip radius

c → length of the hole/crack

cσ σ 0max 2

0

max

σ

A circular hole has a stress concentration factor of 3 [ = 3].

From Inglis’s formula it is seen that the ratio of crack length to crack tip radius is important

and not just the length of the crack.

hole crack

Sharper the crack, higher the stress concentration.

For sharp cracks

= c

For a circular hole

c

cσ σ 210max

0max 3σ σ

One way of understanding this formula is that if maxexceeds t (the theoretical fracture stress), then thematerial fails.

This is in spite of the fact that the applied stress is ofmuch lower magnitude than the theoretical fracture

stress.


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E

cohesive

ca

E f

04

For a crack to propagate the crack-tip stresses have to do work to break the bonds at the

crack-tip. This implies that the ‘cohesive energy’ has to be overcome.

If there is no plastic deformation or any other mechanism of dissipation of energy, the work

done (energy) appears as the surface energy (of the crack faces).

The fracture stress (f ) (which is the ‘far field’ applied stress) can be computed using thisapproach. Note that the fracture stress is of the order of E (i.e. in GPa).

f → fracture stress (applied “far-field”)

→ crack tip radius

c → length of the crack

a0 → Interatomic spacing


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Griffith’s criterion for brittle crack propagation

We have noted that the crack length does not appear ‘independently’ (of the crack tip radius)

in Inglis’s formula. Intuitively we can feel that longer crack must be more deleterious.

Another point noteworthy in Inglis’s approach is the implicit assumption that sufficient

energy is available in the elastic body to do work to propagate the crack. (‘What if there isinsufficient energy?’) (‘What if there is no crack in the body?’). Also, intuitively we canunderstand that the energy (which is the elastic energy stored in the body) should beavailable in the proximity of the crack tip (i.e. energy available far away from the crack tipis of no use!).

Keeping some of these factors in view, Griffith proposed conditions for crack propagation:(i) bonds at the crack tip must be stressed to the point of failure (as in Inglis’s criterion),(ii) the amount of strain energy released (by the ‘slight’ unloading of the body due to crackextension) must be greater than or equal to the surface energy of the crack faces created.

The second condition can be written as:

dc

dU

dc

dU s

Us

→ strain energy

U → surface energy(Energy per unit area: [J/m2])

dc → (‘infinitesimal’) increase in the

length of the crack (‘c’ is the crack length)

We look at the formulae for U s and U next.

Essentially this is like energy balance (with the ‘=‘ sign) → the surface energy for the extended

crack faces comes from the elastically stored energy (in the fixed displacement case)


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The strain energy released on the introduction of a very narrow elliptical double ended

crack of length ‘2c’ in a infinite plate of unit width (depth), under an uniform stress a isgiven by the formula as below.

E U U U a

crack withcrack without

22

s

c

UenergyelasticinReduction

This is because the body with the crack has a lower elasticenergy stored in it as compared to the body without the crack(additionally, the body with the crack is less stiffer). Also, theassumption is that the introduction of a crack does not alterthe far-field stresses (or the load bearing area significantly).

Notes: The units of Us is [J/m] (Joules per meter depth of the crack→ asthis is a through crack). Though Us has a symbol of energy, it is actually a difference

between two energies(i.e. two states of a body→ one with a crack and one without). Half crack length ‘c’ appears in the formula.

E is assumed constant in the process (the apparent modulus will decrease

slightly). a is the ‘far field’ stress (this may result from displacements

rather than from applied forces see note later).

Should be written with a vesign if U = (Ufinal Uinitial)

For now we assume that these stressesarise out of ‘applied’ displacements


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The computation of the actual energy released is more involved and is given by the formulaas noted before:

The formula for Us can be appreciated by considering the energy released from a circularregion of diameter 2c as in the figure below. (The region is cylindrical in 3D).

The energy released is:

22

regioncircular s c21UregioncircularafromreleasedenergyElastic

E a

E

a

22

scU

Energy released from this circular regionis given by the formula (1) as above

(not a true value, but to get a feel of the predominant region involved).

(1)

For a body in plane strain condition (i.e. ~ thick in the z-direction, into the plane of the page), E is replaced with E/(12):

)1(c

U 2

22

s

E a

Plane stress condition

Plane strain condition

E

a

2

s c2

c

U

Hence

As plane strain is more severe on the

material it is better to do experiments in plane strain condition.


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The surface energy of the crack of length 2c & unit width/depth is:

cγU f 4energysurfaceFracture

This is the difference in the energy between a body with a crack and one without a crack.

As pointed out before, the surface energy is the fracture surface energy and not just the

surface free energy. The origin of this energy is contributions from dissipative mechanisms

like plastic deformation, micro-cracking & phase transformation, in addition to the energy

of the ‘broken bonds’.

The units are Joules per meter depth of the body: [J/m].

[J/m]

Important note

The “Griffith experiment” is easily understood in displacement control mode (i.e. apply aconstant displacement and ‘see’ what happens to the crack) and is more difficult to

comprehend it in the force control mode (by applying constant ‘far -field’ forces).

In force control mode, the forces do work on the system and hence the ‘energy accounting’

process is more involved.

Hence, it is better to visualize as arising from ‘far field’ applied displacements.

c f

2c

U


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Now we have the formulae for Us & U (which are required to write down the Griffith’s condition):

dc

dU

dc

dU s f

a

E

2

c 2

LHS increases linearly with c, while RHS is constant.

The ‘equal to’ (=) represents the bare minimum requirement (i.e. the critical condition) →

the minimum crack size, which will propagate with a ‘balance’ in energy (i.e. between

elastic energy released due to crack extension and the penalty in terms of the fracture

surface energy).

The critical crack size (c*): (Note that ‘c’ is half the crack length internal )

A crack below this critical size will not propagate under a constant stress a.

Weather a crack of size greater than or equal to c* will propagate will depend on the Inglis

condition being satisfied at the crack-tip.

This stress a now becomes the fracture stress (f )→ cracks of length c* will grow(unstably) if the stress exceeds f (= a)

2* 2

a

f E c

* f

c

E 2

At constant c (= c*)

when exceeds f then specimen failsGriffith )1(c

E2

2*

f

Plane strain conditions

E

a

2

s c2

c

U

c f

2

c

U

A lt t f d t di th G iffith’ it i ( b d) th h ll I


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E

a f

22c c4Ucrack aof onintroductitheonenergyinChange

c →

U

→

0*

cdc

U d

*c

0c

0

0

An alternate way of understanding the Griffith’s criterion (energy based), though personally I

prefer the previous method

cγU f 4 E a

22

s cU

This change in energy (U) should be negative with an increase in cracklength (or at worst equal to zero). I.e. (dU/dc) ≤ 0.

At c* the slope of U vs c curve is zero [(dU/dc)c* = 0]. This is a pointof unstable equilibrium.

With increasing stress the value of c* decreases (as expected→ more

elastic strain energy stored in the material).

Stablecracks Unstable cracks

Equations for ready reference

Negative slope

Positive slope

c → U →

*

1c*

2c

iffi h li i i


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Griffith versus Inglis criteria

ca

E f

04

Inglis

* f c

E 2

Griffith

result samethe givecriterion Inglisand sGriffith' 8a

If 0

03a Griffith's and Inglis criterion give the same result

the 'Dieter' cross-over criterion

If

2

f

* E 2 c

a

E c

f

20

*

4

For very sharp cracks, the available elastic energy near the crack-tip, will determine if thecrack will grow.

On the other hand if available energy is sufficient, then the ‘sharpness’ of the crack -tip will

determine if the crack will grow.

A sharp crack is limited by availability of energy, while a blunt crack is limited by stress concentration.


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‘Modern’ Fracture Mechanics

G.R. Irwin[1]

Stress Intensity Factor (K)

Material Parameter Fracture Toughness (K C)

Energy Release Rate (G)

Material Parameter Critical Energy Release Rate (GC) J-integral

Material Parameter: JC

[1] G.R. Irwin, “Fracture Dynamics”, in: “Fracture of Metals”, ASM, Cleaveland, OH, 1948, pp.147 -166.[2] G.R. Irwin, “Analysis of stresses and strains near the end of a crack traversing a plate, J. Appl. Mech 24 (1957) 361-364.

F t M h i


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Historically (in the ‘old times’ ~1910-20) fracture was studied using the Inglis and Griffith

criteria.

The birth of fracture mechanics (~1950+) led to the concepts of stress intensity factor (K)

and energy release rate (G). Due to Irwin and others.

Fracture Mechanics

C t f E R l R t (G)


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G is defined as the total potential energy () decrease during unit crack extension (dc):

Concept of Energy Release Rate (G)

dc

d G

The potential energy is a difficult quantity to visualize. In the absence of external

tractions (i.e. only displacement boundary conditions are imposed ), the potentialenergy is equal to the strain energy stored: = Us.*

* It is better to understand the basics of fracture with fixed boundary conditions (without any surface tractions).

dc

dU G s With displacement boundary conditions only

Crack growth occurs if G exceeds (or at least equal to) a critical value GC:

C GG For perfectly brittle solids: GC = 2 f (i.e. this is equivalent to Griffith’s criterion).


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Stress fields at crack tips


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Stress fields at crack tips

For a body subjected far field biaxial stress 0, with a double ended crack of length 2c, thestress state is given by (this is mode-I loading):

23

21

22

SinSinCos

r K I xx

2

3

21

22

SinSinCos

r

K I yy

2

3

222

SinSinCos

r

K I xy

Note the inverse square root (of r) singularity at the crack tip. The intensity of the

singularity is captured by K I (the Stress Intensity Factor ). I.e. K I is the scaling factor for the singularity. As no material can withstand infinite stresses (in ductile materials plasticity will intervene),

clearly the solutions are not valid exactly at (& ‘very near’) the crack tip.

At = 0 and r → the stresses (xx & yy) should tend to 0. This is not the case, as seenfrom the equations ((1) & (2)). This implies that the equations should be used only close to

crack tip (with little errors) or additional terms must be used.

(1)

(2)

(3)

Fig.1

U d t di th t fi ld ti


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Understanding the stress field equation

2

3

21

22

SinSinCos

r

K I xx

r

f K I xx

2

)(→

cY K I 0

‘Shape factor’ related to ‘Geometry’Indicates mode I ‘loading’

Half the crack length

“K I (the Stress Intensity Factor ) quantifies the magnitude of the effect of stress singularity at

the crack tip”[1].

Quadrupling the crack length is equivalent to doubling the stress ‘applied’. Hence, K

captures the combined effect of crack length and loading. The remaining part in equation(1)is purely the location of a point in (r, ) coordinates (where the stress has to be computed).

Note that there is no crack tip radius () in the equation! The assumptions used in thederivation of equations (1-3) are: = 0, infinite body, biaxial loading.

The factor ‘Y’ is considered in the next page.

[1] Anthony C. Fischer-Cripps, “Introduction to Contact Mechanics”, Springer, 2007.

(1)

),( r f K I xx

The Shape factor (Y)


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The Shape factor (Y)

It is obvious that the geometry of the crack and its relation to the body will play an

important role on its effect on fracture.

The factor Y depends on the geometry of the specimen with the crack.

Y=1 for the body considered in Fig.1 (double ended crack in a infinite body). Y=1.12 for a surface crack. The value of Y is larger (by 12%) for a surface crack as

additional strain energy is released (in the region marked dark grey shade in the figure below), due to the presence of the free surface.

Y=2/ for a embedded penny shaped crack.

Y=0.713 for a surface half-penny crack.

Summary of Fracture Criteria


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Summary of Fracture Criteria

Criterion named after &[important quantities]

Comments Fracture occurs if Relevant formulae

Inglis Involves crack tip radius

Griffith Involves crack length

Irwin [K] Concept of stress intensity factor.

K I > K IC

(in mode I)

- [G]Energy release rate based. Same as K based

criterion for elastic bodies.

J-integral

Region of K Dominance


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The crack tip fields consists of two parts: (i) singular part (which blow up near the crack tip)

and (ii) the non-singular part.

The region near the crack tip, where the singular part can describe the stress fields is the K-

Dominance region. This is the region where the stress intensity factor can be used tocharacterize the crack tip stress fields.

Region of K-Dominance

Fracture Toughness (Irwins’s K Based)


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One of the important goals of fracture mechanics is to derive a material parameter, whichcharacterizes cracks in a material. This will be akin to yield stress (y) in a uniaxial tensiontest (i.e. y is the critical value of stress, which if exceeded ( y) then yielding occurs).

The criterion for fracture in mode-I can be written as:

Fracture Toughness (Irwins s K - Based)

IC I K K Where, K IC is the critical value of stress intensity factor (K) and is known

as Fracture Toughness

K IC is a material property (like yield stress) and can be determined for different materialsusing standard testing methods. K

IC

is a microstructure sensitive property.

The focus here is the ‘local’ crack tip region and not ‘global’, as in the case of Griffith’s

approach.

All the restrictions/assumptions on K will apply to K IC: (i) material has a liner elastic behaviour (i.e. no plastic deformation or other non-linear behaviour), (ii) inverse square rootsingularity exists at crack tip (eq. (1)), (iii) the K-dominance region characterizes the crack

tip.

r f K I xx

2)( (1)

Fracture Toughness* (K ) for some typical materials [1]


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Material K IC [MPam]**

Cast Iron 33

Low carbon steel 77

Stainless steel 220

Al alloy 2024-T3 33

Al alloy 7075-T6 28

Ti-6Al-4V 55

Inconel 600 (Ni based alloy) 110

* We have already noted that fracture toughness is a microstructure sensitive property and hence to get ‘true’ value the

microstructure has to be specified.

** Note the strange units for fracture toughness!

[1] Fracture Mechanics, C.T. Sun & Z.-H. Jin, Academic Press, Oxford (2012).

Fracture Toughness (K IC) for some typical materials[ ]

Is K really a material property like ?d h k


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Is K IC really a material property like y?Funda Check

Ideally, we would like K IC (in mode-I loading) (K IIC & K IIIC will be the corresponding

material properties under other modes of loading$) to be a material property, independent of

the geometry of the specimen*. In reality, K IC depends on the specimen geometry andloading conditions.

The value K IC is especially sensitive to the thickness of the specimen. A thick specimen

represents a state that is closer to plane strain condition, which tends to suppress plastic

deformation and hence promotes crack growth (i.e. the experimentally determined value of

K IC

will be lower for a body in plane strain condition). On the other hand, if the specimen is

thin (small value ‘t’ in the figure), plastic deformation can take place and hence the

measured K IC will be higher (in this case if the extent of plastic deformation is large then K I will no longer be a parameter which characterizes the crack tip accurately).

$ Without reference to mode we can call it K C.

* E.g. Young’s modulus is a material property independent of the geometry of the specimen, while stiffness is the equivalent ‘specimen geometrydependent’ property..

To use K IC as a design parameter, we have to use its ‘conservative

value’. Hence, a minimum thickness is prescribed in the standard

sample for the determination of fracture toughness.

This implies that K IC is the value determined from ‘plane strain tests’.

I seem totally messed up with respect to the proliferation of fracture criteria!Q & A


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I seem totally messed up with respect to the proliferation of fracture criteria!How do I understand all this?

Essentially there are two approaches: global (energy based) and local (stress based).

For linear elastic materials the energy and stress field approaches can be considered

equivalent.

Q & A


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Crack propagation in ductile materials


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In brittle materials the maximum crack opening stress (mode-I) is at the crack tip and hencecrack propagation involves breaking of bonds at the crack tip.

In ductile materials the maximum of stress lies a little ahead of the crack tip.

In this scenario the mechanism of cracking changes.

Void(s) nucleate in the region of highest stress (usually at some heterogeneity).

The region between the void and the crack tip acts like a tensile specimen andmicrovoids form in this region. Crack extension occurs by the coalescence ofthese micro-voids.

p p g

What happens to a ‘crack’ in a ductile material?F d Ch k


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What happens to a crack in a ductile material?Funda Check

High magnitude of crack tip stresses can cause yielding at the crack tip (plastic

deformation).

This leads to crack tip blunting, which reduces the stress amplification. There develops a zone ahead of the crack tip known as the process zone.

What else can happen at the crack tip due to high stresses?Funda Check

High magnitude of crack tip stresses can cause:

phase transformation (tetragonal to monoclinic phase in Yttria stabilized Zirconia),

Orowan’s modification to the Griffith’s equation to include “plastic energy”


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E cc )( 4U energyin Change

22

p s

*

p s

f c

E )( 2

Orowan s modification to the Griffith s equation to include plastic energy

232 )1010(~

)21(~

J/m

J/m

p

2

s

*

p

f c

E 2


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Ductile – brittle transition


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Certain materials which are ductile at a given temperature (say room temperature), become

brittle at lower temperatures. The temperature at which this happens is terms as theDuctileBrittle Transition Temperature (DBTT).

As obvious, DBT can cause problems in components, which operate in ambient and lowtemperature conditions.

Typically the phenomena is reported in polycrystalline materials. Deformation should be

continuous across grain boundary in polycrystals for them to be ductile. This implies that

five independent slip systems should be operative (this is absent in HCP and ionic

materials). This phenomenon (ductile to brittle transition) is not observed in FCC metals (they remain

ductile to low temperatures).

Common BCC metals become brittle at low temperatures (as noted before a decrease in

temperature can be visualized as an increase in strain rate, in terms of the effect on the

mechanical behaviour).

As we have noted before a ductile material is one which yields before fracture (i.e. its yield

strength is lower in magnitude than its fracture strength).

What causes the ductile to brittle transition phenomenon?


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Both the fracture stress (f ) and the yield stress (y) are temperature dependent. However,as slip is a thermally activated process, the yield stress is a stronger function of temperatureas compared to the fracture stress.

If one looks at the Griffith’s criterion of fracture, f has a slight dependence on temperatureas E increases with decreasing the temperature ( also has a slight temperature dependence,which is ignored here). y on the other hand has a steeper increase with decreasingtemperature.

p

f , y →

y

T →

f

DBTT

DuctileBrittle

Ductile y < f yields before fracture

Brittle y > f fractures before yielding

* f c

E 2

Griffith’s criterion


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ca

E f

04

Inglis


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f , y →

y (BCC)

T →

f

DBTT

y (FCC)

No DBTT

Griffith versus Hall-Petch


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Griffith versus Hall Petch

* f

c

E 2

d

k i y

Griffith Hall-Petch

*

'1

c

k

c

E 2

* f


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f , y →

y

d-½ →

DBT

T1

T2

T1 T2f

Grain size dependence of DBTT

Finer sizeLarge size

Finer grain size has higher DBTT better

T1T2 >


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f , y →

y

d-½ →

DBT

T1

T2

T1 f

Grain size dependence of DBTT- simplified version - f f(T)

Finer size

Finer grain size has lower DBTT better

T1T2 >

Protection against brittle fracture

E 2


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g

Lower value of surface energy () implies a lower fracture stress (f ) done by chemicaladsorption of molecules on the crack surfaces.

Removal of surface cracks* etching of glass (followed by resin cover).

The best method is by introducing residual compressive stresses on the surface. This can bedone by: Surface of molten glass solidified by cold air followed by solidification of the bulk. Theshrinkage of the inner glass leads to compressive stress on the surface (this is called tempered glass)→ this way the fracture strength can be increased 2-3 times. Ion exchange method → smaller cations like Na+ in sodium silicate glass are replaced by

larger cations like K + on the surface of glass → higher compressive stresses than tempering. Shot peening (surface of metal is impacted by round particles to introduce residualcompressive stress). Carburizing and Nitriding (these processes are used to increase the hardness, butadditionally give a benefit in terms of the residual compressive stress introduced). Pre-stressed concrete (concrete is cast around pre-tensioned steel cables or bars thetension is then released thus introducing compressive load on the concrete).

Cracks developed during grinding of ceramics extend upto one grain use fine grainedceramics (grain size ~ 0.1 m).

Avoid brittle continuous phase along the grain boundaries → path for intergranular fracture

(e.g. iron sulphide film along grain boundaries in steels → Mn added to steel to form

spherical manganese sulphide).

* f c

* As surface cracks are more deleterious

check


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Conditions of fracture

Torsion

Fatigue

Tension

Creep

Low temperature Brittle fracture

Temper embrittlement

Hydrogen embrittlement

Why do we need a large ductility (say more than 10% tensile elongation)Funda Check


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material, while ‘never’ actually in service component is going to see/need suchlarge plastic deformation (without the component being classified as ‘failed’).

Funda Check

Let us take a gear wheel for an example. The matching tolerances between gears are so

small that this kind of plastic deformation is clearly not acceptable. In the case of the case carburized gear wheel, the surface is made hard and the interior is

kept ductile (and tough).

The reason we need such high values of ductility is so that the crack tip gets blunted and the

crack tip stress values are reduced (thus avoiding crack propagation).

E2*


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→

c

→

Fracture

stable

E 2 c

2

*

*c

00

Rajesh Prasad’s Diagrams Validity domains for brittle fracture criteria


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Sharpest possible crack Approximate border for changeover of criterion

→

c

→

a0 3a0

Validityregion

for Energy

criterionGriffith

Validityregion

for Stresscriterion

Inglis

Sharp

cracks

Blunt

cracks

> c

= c


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→

c

→

a0

c*

Safety regions applying Griffith’s criterion alone

Unsafe

Safe

2

f

* E 2 c


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UnsafeSafe

→

c

→

a0

Safety regions applying Inglis’s criterion alone

a

E c

f

2

0

*

4


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→

c

→

a0

c*

3a0

Griffith safeInglis unsafe safe

Griffith unsafeInglis safe safe

Griffith safeInglis unsafe unsafe

Griffith unsafe

Inglis unsafe unsafe

Griffith safeInglis safe safe

Role of Environment in Fracture


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Role of Environment in Fracture

Stress Corrosion Cracking

Hydrogen Embrittlement

Stress Corrosion Cracking (SCC)


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In stress corrosion cracking the presence of a chemical species can enhance crack

propagation and reduce fracture stress. This phenomenon can lead to sudden failure ofductile metals, especially at high temperatures. The interplay between stress and corrosion is important here.

The chemical agent is one which is normally corrosive to the metal/alloy* involved. Certaincombinations of metals and chemicals can lead to disastrous effects (i.e. the good news is that not allcombinations are that bad).

Similar to a critical value of the stress intensity factor (K IC) in normal fracture mechanics,

we can define a critical stress intensity factor in the presence of a corrosive environment (at the

crack tip) (K ISCC). This value as seen from the table below can be much lower than K IC.

Severe accidents like the explosion of boilers, rupture of

gas pipes, etc. have happened due to this phenomenon.

* Metals are considered here, although other materials are also prone to such effects.

Sudden crack growth on exceeding K ISCC

Unlike K IC, K ISCC is not a pure material parameter and is

affected by environmental variables (hence for each environment-

material pair the appropriate K ISCC value has to be used).


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AlloyK IC

(MN/m3/2)SCC

environmentK ISCC

(MN/m3/2)

13Cr steel 60 3% NaCl 12

18Cr-8Ni 200 42% MgCl2 10

Cu-30Zn 200 NH4OH, pH7 1

Al-3Mg-7Zn 25 Aqueous halides 5

Ti-6Al-1V 60 0.6M KCl 20

http://en.wikipedia.org/wiki/Stress_corrosion_cracking

Another related phenomenon, which can be classified under the broad ambit of SCC is

http://en.wikipedia.org/wiki/Stress_corrosion_crackinghttp://en.wikipedia.org/wiki/Stress_corrosion_cracking


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hydrogen embrittlement.

Hydrogen may be introduced into the material during processing (welding, pickling,

electroplating, etc.) or in service (from nuclear reactors, corrosive environments, etc.).


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Q & A What are the characteristics of brittle fracture


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Extreme case scenario is considered here:

Cracks are sharp & no crack tip blunting.

No energy spent in plastic deformation at the crack tip. Fracture surfaces are flat.

Q & A What is the difference between plane stress and plane strain as far as fracture goes?


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C


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END

Ductile fracture


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Ductile fracture →

► Crack tip blunting by plastic deformation at tip► Energy spent in plastic deformation at the crack tip

→

y

r →

→

y

r →Sharp crack Blunted crack

Schematic

Documents

Chapter 9d Fracture