Defects in Solids• 0-D or point defects

– vacancies, interstitials, etc.– control mass diffusion

• 1-D or linear defects– dislocations– control deformation processes

• 2-D or planar defects– grain boundaries, surfaces, interfaces, – heterophase boundaries

• 3-D or volume defects– voids, secondary components (phases)

Surface Tension as a Force

FL

surface tension, g energy / area

to increase liquid area, energy is required

we must do work on the system

dwrev = Fdx = gdA A = 2Lx

x dx

dA = 2Ldx

Fdx = 2gLdx F = 2gL

Force of surface tension acts in a direction parallel to surface

Imaging Grain Boundaries

external surface

grains in the solid

if smooth, difficult to observe

surface tension smooth is not thermodynamically preferred

thermal anneal or thermal ‘etch’ to equilibrate

grain 1 grain 2

f

gSS

gSVgSV

gSS = 2gSVcos(f /2)

balance of forces

Defects in Solids• 0-D or point defects

– vacancies, interstitials, etc.– control mass diffusion

• 1-D or linear defects– 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 properties – yield, metals)

(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

B

• At grain boundaries– Second phase concentrated at triple

contacts of host grain boundaries– Typical when liquid phase forms at

high temperature

liquid / amorphous

grain 1grain 3

grain 2

Balance of forces

interphase boundary

gSS

gLS gLS

f

gSS = 2gSLcos(f/2)

• Pores– 2nd phase is a void– increases scattering– thermal insulation– white, not transparent

Volume Defects and Mechanics• A secondary (different) material: “phase”

– in metals: secondary phases tend to pin dislocations• Pores

– in ceramics: tend to be source of failure

s = F/A

e = DL/L

ceramic

metal

Mechanical Behavior

sy

sfrac

Y

Y

Y – from chemical bonds

sy – due to dislocation glide

sy (obs) << sy (theo)

sfrac – due to volume defects

sfrac (obs) << sfrac (theo)

F

F

L

x

“graceful” failure

“catastrophic” failure

Evaluate sfrac(theoretical)

F = dE/dR

attra

ctiv

ere

puls

ive

F

RR0

ER (interatomic distance)

E0

R0

bond energy curve

bond force curve

s = F/A

theofracs

approximate s as sinusoidal

x

ths

2πsinthxs s

R0 ~ a0

0

xYa

linear region: Ys e

2π~ thxs s

0

xYa

02πthY

as

fracture plane

F

F

???

/2

x = 0

simultaneous failure

0

RYR

D

Evaluate sfrac(theoretical)

02πthY

as 1. ~ ao 2πth

Ys

2. Obtain by equating mechanical energy (work) of creating two surfaces to their surface energy

a0

ths

/2

x = 0

s

x

E Fdx /E area dxs2

0

2π/ sinthxwork area dx

s

πths

surface energy / area of fracture = 2g2 π

th

gs

2 π

2πtho th

Yagss

12

gs

tho

Ya

Griffith’s equation If some plastic deformation occurs:

geff = gsurf + gplastic

Evaluate sfrac(observed)1

2

0

~ gs s

obs theofrac frac

Ya

Stress concentration at crack tips

12

2tip appcs s

1122

2 appo

c Yags

Why??

2c only this region of the material supports the load

can show:radius of curvature

take fracture to occur when: s s theotip frac

12

4appo

Ya c

g s

obsfracs ao

12

4obsfrac

Yc

gs

in general: ½ s obsfrac c

measured fracture stress is not an “inherent” material property

s = F/A

inte

rnal

forc

e lin

es

atomically sharp crack tip

12s stip appc

obsfracs

Evaluate sfrac(observed)

Alternative derivation: again, consider energy balance

0( )E c E

+ surface energy

take fracture to occur when: c > c*obsfracs

122

πobsfrac

Ycgs

as before:

½obsfrac cs

measured fracture stress is not an “inherent” material property

= initial energy

- released strain energy

22π0 4E cc Y

sg

E(c)

c

energy per unit thickness

crack length

crack energy

c*

2 2π cYs

4cg

2c

Fracture Behavior• In general:

@ failure:

custom:

not @ failure:K, KC units: pressure (length)½

in practice, need to specify geometry of the experimentshear vs. tension, etc. geometric constant

characterization: put in a crack of known length and defined geometry

½,geometrytip app appf c kcs s s

½frac fractip app kcs s

½

theofracobs

frac kcs

s

½ ½(π ) (π )

theofracobs C

fracK

k c cs

s fracture toughness

stress intensity factorcritical stress intensity

indep. of geometry depends on crack length

depends on geometry

theofrac

ks

To strengthen ceramics, pay attention to cracks

½)π( cK appls

½(π )obsC fracK cs

KI Stress Intensity Factors

Chiang, Bernie, and Kingery, “Physical Ceramics: Principles for Ceramic Science & Engineering” Wiley 1997

Crack-Loading Modes

Courtney, “Mechanical Behavior of Materials,” McGraw-Hill 2000

½(π )applK csI

molten potassium

salt

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

s

tension

compression

stension

compression

NaO*SiO2

K

Na

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 transitionincrease in volume upon transitionDV places compressive stress on crack (closes it)

Mechanical Properties• Elastic properties

– depend on chemical bonding, not so sensitive to slight variations in composition, processing

• Yield stress (metals)– can be manipulated by processing– fairly reproducible

• Fracture stress (ceramics)– an almost meaningless property– depends on details of crack/pore distribution– achieving reproducibility is a major effort