Transcript lecture 17 powerpoint

Defects in Solids
• 0-D or point defects 
– vacancies, interstitials, etc.
– control mass diffusion
• 1-D or linear defects 
(mechanical properties – yield, metals)
– dislocations
– control deformation processes
• 2-D or planar defects 
– grain boundaries, surfaces, interfaces,
– heterophase boundaries
• 3-D or volume defects
– voids, secondary components (phases)
(mechanical props – fracture, ceramics)
Volume Defects / Heterophase Boundaries
• Composites
– Two or more distinct types of materials, “phases”
– Boundary between them is a heterophase interface
A
• At grain boundaries
– Second phase concentrated at triple
contacts of host grain boundaries
– Typical when liquid phase forms at
high temperature
B
interphase boundary
liquid / amorphous
grain 2
• Pores
grain 1
Balance of forces
gSS = 2gSLcos(f/2)
f
gLS
grain 3
gLS
gSS
–
–
–
–
2nd phase is a void
increases scattering
thermal insulation
white, not transparent
Volume Defects and Mechanics
F
• A secondary (different) material: “phase”
L
– in metals: secondary phases tend to pin dislocations
• Pores
– in ceramics: tend to be source of failure
Mechanical Behavior
sfrac
s = F/A
Y – from chemical bonds
ceramic
x
Y
sy
“catastrophic”
failure
metal
“graceful” failure
Y
e = DL/L
sy – due to dislocation glide
sy (obs) << sy (theo)
sfrac – due to volume defects
sfrac (obs) << sfrac (theo)
F
Evaluate sfrac(theoretical)
F
simultaneous failure
bond energy curve
fracture plane
E
R (interatomic distance)
F
R0
F = dE/dR
E0
approximate s as sinusoidal
 2πx 

  
attractive
s  s th sin 
 s th
s theo
frac
F
repulsive
s = F/A
R0
/2
x=0
bond force curve
 x
 DR 
linear region: s  Y e  Y 
 Y  
R
 0 
 a0 
Rx
s ~ s th
R0 ~ a0
 s th 
2πx

Y 
2π a0
 x
Y  
 a0 
???
Evaluate sfrac(theoretical)
Y 
s th 
2π a0
1.  ~ ao
Y
 s th 
2π
2. Obtain  by equating mechanical energy (work) of creating
two surfaces to their surface energy
s th
s
E   Fdx
E / area   s dx

work / area   2 s
0
/2
a0
x

x=0
surface energy / area of fracture = 2g
 gY 
 s th  

a
 o 
1
2
Griffith’s equation
 
2g π
s th
 2πx 
sin
th

 dx
  
s th
π
 s th 
Y 2g π
2π aos th
If some plastic
g = gsurf + gplastic
deformation occurs: eff
Evaluate sfrac(observed)
s
obs
frac
 s
th
frac
 gY 
~

a
 0 
1
2
Why??
internal force lines
s = F/A
2c
can show: s tip
only this region
of the material
supports the load
c
 2s app  

 s
obs
frac
 gY 


 4c 
Stress concentration at crack tips
1
c
 2s app  

take fracture to occur when:
1
2
 gY 


a
 o 
1
s obs
frac
2
 s app
2
in general:
 gY  


4
a
c
 o 
ao
1
2
1
2
radius of
curvature
s tip  s thfrac
atomically sharp
crack tip
½
s obs

c
frac
measured fracture stress is not an “inherent” material property
Evaluate sfrac(observed)
Alternative derivation: again, consider energy balance
crack
energy
2c
E(c)
= initial + surface
energy
energy
- released
strain energy
πs 2c 2
E (c)  E0  4cg 
Y
energy
per unit
thickness
c*
c
take fracture to occur when: c > c*
crack length
 s
obs
frac
 2g Y 


π
c


1
E
2πs 2 c
 0  4g 
c
Y
2
as before:
s obs
frac
½
s obs

c
frac
measured fracture stress is not an “inherent” material property
Mechanical Properties
• Elastic properties
– depend on chemical bonding, not so sensitive
to slight variations in composition, processing
• Yield stress (metals)
– depends on details of processing
– fairly reproducible
• Fracture stress (ceramics)
– an almost meaningless property
– depends on details of crack/pore distribution
– achieving reproducibility is a major effort
Fracture Behavior
s tip  s app f  c,geometry   s app kc½
• In general:
@ failure:
frac
s tipfrac  s app
kc½
indep. of geometry
custom:
s
depends on
crack length
obs
frac
 s obs
frac 
s theo
frac
kc½
depends on
geometry
s theo
frac
s theo
frac
fracture
toughness
KC


½
k (πc)
(πc)½
k
½
KC  s obs
frac (πc)
critical stress intensity
not @ failure:
K  s appl (πc)½
stress intensity factor
K, KC units: pressure  (length)½
in practice, need to specify geometry of the experiment
shear vs. tension, etc.  geometric constant
characterization: put in a crack of known length and defined geometry
To strengthen ceramics, pay attention to cracks
Strengthening of Ceramics
•
•
•
•
Process to eliminate cracks (internal)
Polish to eliminate surface cracks
Blunt crack tip
Anneal (heat treat) to eliminate randomly
distributed internal stresses
• Quench (a silicate glass) to induce
compressive stress on surface
• Ion exchange to induce surface compressive
stress
molten
potassium
salt
tension
NaO*SiO2
s
Na
K
compression
tension
s
compression
once crack penetrates
compressive region,
material shatters
explosively
Strengthening of Ceramics
• Transformation toughening
• Cool ZrO2: cubic  tetragonal  monoclinic
• Modify with CaO:
cubic  tetragonal  monoclinic + cubic
• Rapid cooling: tetragonal  monoclinic is slow
obtain tetragonal + cubic
cubic
tetragonal
crack catalyzes tetragonal  monoclinic transition
increase in volume upon transition
DV places compressive stress on crack (closes it)