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FRACTURE



Brittle Fracture



Ductile to Brittle transition

Fracture Mechanics


T.L. Anderson



CRC Press, Boca Raton, USA (1995)

Breaking

of

Liberty Ships

Cold waters

Welding instead of riveting

High sulphur in steel

Residual stress

Continuity of the structure

Microcracks

Fracture

Brittle

Ductile

Factors affecting fracture

Strain rate

State of stress

Temperature

Behaviour described

Terms Used

Crystallographic mode

Shear

Cleavage

Appearance of Fracture surface

Fibrous

Granular / bright

Strain to fracture

Ductile

Brittle

Path

Transgranular

Intergranular

Conditions of fracture

Torsion

Fatigue

Tension

Creep

Low temperature Brittle fracture

Temper embrittlement

Hydrogen embrittlement



Brittle fracture



Little or no deformation



Observed in single crystals and polycrystals



Have been observed in BCC and HCP metals but not in FCC metals

Types of failure

Promoted by

High Strain rate

Triaxial state of State of stress

Low Temperature



Shear fracture of ductile single crystals



Not observed in polycrystals

Slip plane



Completely ductile fracture of polycrystals
→ rupture



Very ductile metals like gold and lead behave like this



Ductile fracture of usual polycrystals



Cup and cone fracture



Necking leads to triaxial state of stress



Cracks nucleate at brittle particles
(void formation at the matrix
-
particle


interface)

Theoretical shear strength and cracks



The theoretical shear strength
(to break bonds and cause fracture)


of perfect crystals ~ (E / 6)



Strength of real materials ~ (E / 100 to E /1000)



Tiny cracks are responsible for this



Cracks play the same role in fracture
(of weakening)


as dislocations play for deformation

Applied Force (F)


r


a
0

Cohesive force


E

cohesive



=

2a

a

Characterization of Cracks



Surface or interior



Crack length



Crack orientation with respect to geometry and loading



Crack tip radius

Crack growth and failure



Brittle fracture

Crack growth criteria

Stress based

Energy based



Global



~Thermodynamic



Local



~Kinetic

Griffith

Inglis

For growth of crack

Sufficient stress concentration should

exist at crack tip to break bonds

It should be energetically favorable



Brittle fracture →





cracks are sharp & no crack tip blunting




No energy spent in plastic deformation at the crack tip

Griffith’s criterion for brittle crack propagation



When crack grows

c


4

energy

surface
in

Increase


E

c


energy

elastic

in

Reduction
2
2



E

c


c


4


U

energy

in

Change
2
2







c



U


0


dc
U
d
0


U
*
c
0
c
size
crack

critical
c

*
c



U


*
1
c
0
c
*
2
c


E


2


c
2
*




By some abracadabra

*
f
c

E


2







At constant stress

when c > c
*

by instantaneous

nucleation

then specimen fails


At constant c (= c
*

→ crack length)

when


exceeds

f

then specimen

fails

Griffith



If a crack of length c
*

nucleates “
instantaneously”

then it can grow with


decreasing energy →
sees a energy downhill



On increasing stress the critical crack size decreases





c



Fracture

stable



E


2


c
2
*




*
c

0


0

To derive c
*

we differentiated w.r.t

c

keeping


constant

Stress criterion for crack propagation



Cracks have a sharp tip and lead to stress concentration


0










c
σ
σ
2
1
0
max



0

→ applied stress




max

→ stress at crack tip





→ crack tip radius


c
σ
σ
0
max
2



= c

For a circular hole









c
c
σ
σ
2
1
0
max
0
max
3
σ
σ


E

cohesive




Work done by crack tip stresses to create a crack
(/grow an existing crack)



= Energy of surfaces formed

c
a
E
f
0
4



After lot of approximations

Inglis



a
0

→ Interatomic spacing

Griffith versus Inglis

c
a
E
f
0
4



Inglis

*
f
c

E


2






Griffith

result

same
the

give

criterion

Inglis

and

s
Griffith'

8a

If
0



0
3a Griffith's and Inglis criterion give
the same result
the 'Dieter
' cross-over criterion
If



2
f
*
E


2

c









a

E
c
f









2
0
*
4
Rajesh Prasad’s Diagrams

Validity domains for brittle fracture criteria

Sharpest possible crack

Approximate border for changeover of criterion





c


a
0

3a
0

Validity

region

for

Energy

criterion

Griffith

Validity

region

for

Stress

criterion

Inglis

Sharp

cracks

Blunt

cracks



> c



= c





c


a
0

c
*

Safety regions applying
Griffith’s

criterion alone

Unsafe

Safe

2
f
*
E


2

c




Unsafe

Safe





c


a
0

Safety regions applying
Inglis’s
criterion alone






a

E
c
f









2
0
*
4




c


a
0

c
*

3a
0

Griffith safe

Inglis unsafe



safe

Griffith unsafe

Inglis safe



safe

Griffith safe

Inglis unsafe



unsafe

Griffith unsafe

Inglis unsafe



unsafe

Griffith safe

Inglis safe



safe

Ductile


brittle transition



Deformation should be continuous across grain boundary in polycrystals


for their ductile behaviour


5 independent slip systems required


(absent in HCP and ionic materials)



FCC crystals remain ductile upto 0 K



Common BCC metals become brittle at low temperatures or at v.high


strain rates



Ductile



y

<

f



yields before fracture



Brittle



y

>

f



fractures before yielding


f

,

y




y

T




f

DBTT

Ductile

Brittle

Ductile


yields before fracture

Brittle


fractures before yield

c
a
E
f
0
4



Inglis

*
f
c

E


2






Griffith


f

,

y




y

(BCC)

T




f

DBTT


y

(FCC)

No DBTT

Griffith versus Hall
-
Petch

*
f
c

E


2






d
k
i
y




Griffith

Hall
-
Petch

*
'
1
c
k
c

E


2


*
f






f

,

y




y

d
-
½




DBT

T
1

T
2

T
1

T
2


f

Grain size dependence of DBTT

Finer size

Large size

Finer grain size has
higher

DBTT


better

T
1

T
2

>


f

,

y




y

d
-
½




DBT

T
1

T
2

T
1


f

Grain size dependence of DBTT
-

simplified version
-


f



f(T)


Finer size

Finer grain size has
lower

DBTT


better

T
1

T
2

>

Protection against brittle fracture









f




done by chemical adsorbtion

of molecules


on the crack surfaces



Removal of surface cracks


etching of glass


(followed by resin cover)



Introducing compressive stresses on the surface




Surface of molten glass solidified by cold air followed by



solidification of the bulk
(tempered glass)



→ 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




Carburizing and Nitriding




Pre
-
stressed concrete

*
f
c

E


2








Cracks developed during grinding of ceramics extend upto one grain




use fine grained ceramics (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)



Ductile fracture →





Crack tip blunting by plastic deformation at tip




Energy spent in plastic deformation at the crack tip

Ductile fracture






y

r








y

r



Sharp crack

Blunted crack

Schematic

r

→ distance from the crack tip

E

c


c

)
(

4


U

energy

in

Change
2
2
p
s









*
p
s
f
c

E

)
(

2








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

2
3
2
)
10
10
(
~
)
2
1
(
~
J/m

J/m

p
2
s




*
p
f
c

E


2