# Lecture Recap: 9/21/09

```Lecture Recap: 9/21/09
Dislocation Energy
γ
1
W = E = ∫ τ d γ = τγ
2
0
W
μb 2
=
V
8π 2 r 2
rz
1
= μγ
2
2
rz
Screw
W
2
~ μb
l
W
screw
Edge
W μb
~
l 1 −ν
2
Same dimensions as F, “line tension”
y Edge dislocation always higher energy
{
(1-ν)&lt;1
y Crystals
y
tryy to form long
g screw dislocations
{
Dislocations often zigzag to accommodate screw
http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_5/backbone/r5_2_3.html
∫ ∫
z
r
2
μb2
r
W = l
ln( outer )
4π
r inner
y Energy of dislocation proportional to length
{
=
μb2
8π 2 r
Mixed Dislocations
Stress
σxx
σyy
Edge
Screw
μby
3x2 + y2
−
2 π (1 − ν ) ( x 2 + y 2 ) 2
μby
x2 − y2
2 π (1 − ν ) ( x 2 + y 2 ) 2
σzz
ν (σ
τxyy
μbx
xx
+σ
yy
)
x2 − y2
2 π (1 − ν ) ( x 2 + y 2 ) 2
τxz
0
τyz
0
0
0
0
0
μb
cos Θ
2πr
μb
sin Θ
2πr
Stress Fields
y Dislocations can interact
y Imagine them like charges: similar dislocations
repel, opposites attract
Image of stress fields around two dislocations removed due to copyright restrictions.
http://www.matter.org.uk/matscicdrom/manual/images/image109.gif
Dislocation Motion
y Peach-Koehler Equation
Fx = bxσ xy + byσ yy + bzσ zy
Fy = −(bxσ xx + byσ xy + bzσ xz )
Edge Dislocation
σxx
F
σyy
F=0
σzz
F=0
0
F
τxy
τxz
F=0
τyz
F
0
F=0
Screw Dislocation
σxx
F= 0
σyy
F= 0
Image removed due to copyright restrictions.
σzz
=0
00 Please see the cover of Nature Physics 5
F=
τxy
F=
=00 (April 2009).
F
τxz
F
τyz
*assumes pos Edge, RH Screw
http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_5/illustr/dislocation_3dim.jpg
Dislocation Motion
y Dislocation moves along
gp
plane containing
g b,, t
Edge
Screw
S
S
t
b
MIT OpenCourseWare
http://ocw.mit.edu
3.40J / 22.71J / 3.14 Physical Metallurgy
Fall 2009