Transcript FRACTURE  Brittle Fracture Ductile to Brittle transition

FRACTURE
 Brittle Fracture
 Ductile to Brittle transition
Fracture Mechanics
T.L. Anderson
CRC Press, Boca Raton, USA (1995)
Continuity of the structure
Welding instead of riveting
Breaking
of
Liberty Ships
Residual stress
Microcracks
Cold waters
High sulphur in steel
Ductile
Fracture
Brittle
Temperature
Factors affecting fracture
Strain rate
State of stress
Behaviour described
Terms Used
Crystallographic mode
Shear
Cleavage
Appearance of Fracture surface
Fibrous
Granular / bright
Strain to fracture
Ductile
Brittle
Path
Transgranular
Intergranular
Tension
Torsion
Fatigue
Conditions of fracture
Creep
Low temperature Brittle fracture
Temper embrittlement
Hydrogen embrittlement
Types of failure
Low Temperature
Promoted by
High Strain rate
Triaxial state of State of stress
 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
Slip plane
 Shear fracture of ductile single crystals
 Not observed in polycrystals
 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
Applied Force (F) →
 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
Cohesive force
a0
 cohesive 
r →
E

Characterization of Cracks
2a
=
a
 Surface or interior
 Crack length
 Crack orientation with respect to geometry and loading
 Crack tip radius
Crack growth and failure
 Brittle fracture
Griffith
Energy based
 Global
 ~Thermodynamic
Crack growth criteria
Stress based
Inglis
 Local
 ~Kinetic
It should be energetically favorable
For growth of crack
Sufficient stress concentration should
exist at crack tip to break bonds
 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
Increase in surface energy  4  c
Reduction in elastic energy   c 2
2
E
Change in energy  U  4  c   c 2
2
E
U →
d U
0
dc
U  0
c*
c0
c *  critical crack size
c →
U →
c1* c2*
c0
c →
Griffith
c 
*
2 E
2
By some abracadabra
At constant stress
when c > c* by instantaneous
nucleation then specimen fails
2 E
f 
 c*
At constant c (= c* → crack length)
when  exceeds f then specimen
fails
c 
2 E
2
To derive c* we differentiated w.r.t
c keeping  constant
c →
*
Fracture
c
0
*
stable
0
 →
 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
Stress criterion for crack propagation
 Cracks have a sharp tip and lead to stress concentration
0
σ max

c
 σ 0 1  2




 0 → applied stress
 max → stress at crack tip
  → crack tip radius
For a circular hole
σ max  2σ 0
c

c
σ max  σ 0 1  2 
c


σ max  3σ 0
=c
 cohesive 
E

Work done by crack tip stresses to create a crack (/grow an existing crack)
= Energy of surfaces formed
After lot of approximations
Inglis
f 
E
4a 0 c
 a0 → Interatomic spacing
Griffith versus Inglis
Inglis
Griffith
2 E
f 
 c*
If  
8a0

f 
E
4a 0 c
Griffith' s and Inglis criterion give the same result
If   3a 0 Griffith's and Inglis criterion give the same result
 the 'Dieter' cross-over criterion
c* 
2 E
 2f
 E
c 
4a  2
 0 f
*




c →
Rajesh Prasad’s Diagrams
Validity
region
for
Energy
criterion
Griffith
Validity domains for brittle fracture criteria
Validity
region
for
Stress
criterion
Inglis
=c
Blunt
cracks
Sharp
cracks
>c
a0
Sharpest possible crack
3a0
 →
Approximate border for changeover of criterion
c →
Safety regions applying Griffith’s criterion alone
Unsafe
c 
*
c*
Safe
a0
 →
2 E
 2f
c →
Safety regions applying Inglis’s criterion alone
Unsafe
a0
 E
c 
4a  2
 0 f
Safe
*
 →




c →
Griffith unsafe
Inglis unsafe
 unsafe
c*
Griffith unsafe
Inglis safe
 safe
Griffith safe
Inglis unsafe
 unsafe
Griffith safe
Inglis unsafe
 safe
Griffith safe
Inglis safe
 safe
a0
3a0
 →
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
Griffith
y
2 E
f 
 c*
f , y →
Inglis
f
f 
Ductile
Brittle
T →
DBTT
Brittle  fractures before yield
Ductile  yields before fracture
E
4a 0 c
f , y →
f
y (BCC)
y (FCC)
T →
DBTT
No DBTT
Griffith versus Hall-Petch
Hall-Petch
Griffith
2 E
f 
 c*
f 
2 E

k
 y i 
d
1
k'

*
c
c*
Grain size dependence of DBTT
T1
T2
f
T2
f , y →
y
T1
T2
Large size
Finer size
d-½ →
DBT
Finer grain size has higher DBTT  better
>
T1
Grain size dependence of DBTT- simplified version - f  f(T)
T1
T2
f
f , y →
y
T1
T2
Finer size
d-½ →
DBT
Finer grain size has lower DBTT  better
>
T1
Protection against brittle fracture
 ↓  f ↓  done by chemical adsorbtion of molecules
2 E
f 
on the crack surfaces
 c*
 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
 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
 Ductile fracture →
► Crack tip blunting by plastic deformation at tip
► Energy spent in plastic deformation at the crack tip
y
y
 →
 →
Schematic
Sharp crack
r →
r → distance from the crack tip
Blunted crack
r →
Orowan’s modification to the Griffith’s equation to include “plastic energy”
Change in energy  U  4 (  s   p ) c   c 2
f 
f 
2 ( s   p ) E
c
2 p E
 c*
*
2
E
 s ~ (1  2) J/m 2
 p ~ (10 2  10 3 ) J/m 2