FRACTURE

Brittle Fracture

Ductile to Brittle transition

Fracture MechanicsT.L. Anderson

CRC Press, Boca Raton, USA (1995)

Breakingof

Liberty ShipsCold 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

App

lied

For

ce (

F)

→

r →a0

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 growsc 4 energy surfacein Increase

E c energy elastic in Reduction

22

E c c 4 U energy in Change

22

c →

U →

0dc

Ud

0U

*c0c

sizecrack criticalc *

c →

U →

*1c

0c*2c

Increasing str

ess

E 2 c

2*

By some abracadabra*f c

E 2

At constant stresswhen 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.tc keeping constant

Stress criterion for crack propagation

Cracks have a sharp tip and lead to stress concentration

0

c

σσ 210max

0 → applied stress

max → stress at crack tip

→ crack tip radius

c

σσ 0max 2

= c

For a circular hole

c

cσσ 210max

0max 3σσ

E

cohesive

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

= Energy of surfaces formed

ca

Ef

04

After lot of approximations

Inglis

a0 → Interatomic spacing

Griffith versus Inglis

ca

Ef

04

Inglis

*f c

E 2

Griffith

result samethe give criterion Inglis and sGriffith' 8a

If 0

0 3a Griffith's and Inglis criterion give the same result

the 'Dieter' cross-over criterion

If

2f

* E 2 c

a

Ec

f

20

*

4

Rajesh Prasad’s Diagrams Validity domains for brittle fracture criteria

Sharpest possible crack Approximate border for changeover of criterion

→

c →

a0 3a0

Validityregion

forEnergycriterion

Griffith

Validityregion

forStresscriterion

Inglis

Sharp cracks

Blunt cracks

> c

= c

→

c →

a0

c*

Safety regions applying Griffith’s criterion alone

Unsafe

Safe

2f

* E 2 c

UnsafeSafe

→

c →

a0

Safety regions applying Inglis’s criterion alone

a

Ec

f

20

*

4

→

c →

a0

c*

3a0

Griffith safeInglis unsafe safe

Griffith unsafeInglis safe safe

Griffith safeInglis unsafe unsafe

Griffith unsafeInglis unsafe unsafe

Griffith safeInglis safe safe

Ductile – brittle transition

Deformation should be continuous across grain boundary in polycrystalsfor 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

DuctileBrittle

Ductile yields before fractureBrittle fractures before yield

ca

Ef

04

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

kiy

Griffith Hall-Petch

*

'1

c

k

c

E 2

*f

f ,

y →

y

d-½ →

DBT

T1

T2

T1T2f

Grain size dependence of DBTT

Finer sizeLarge size

Finer grain size has higher DBTT better

T1T2 >

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

↓ f ↓ done by chemical adsorbtion of moleculeson 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 sodiumsilicate 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

22

ps

*

psf c

E )( 2

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

232 )1010(~

)21(~

J/m

J/m

p

2s

*

pf c

E 2