材料科学基础
Fundamental of Materials
Prof,Tian Min Bo
Tel,62795426, 62772851
E-mail,tmb@mail.tsinghua.edu.cn
Department of Material Science and Engineering
Tsinghua University,Beijing 100084
Lesson twenty-one
§ 5.10 Forces on dislocation
? Ⅰ.Introduction
1,Definition:
is a fictitious force which is supposed to be responsible for the
motion of dislocation.
∥
Criterion:
The work done on crystal by stress must be equal to that
done on dislocation by force F.
2,Principle of derivation of,
Macroscopic work＝ microscopic work
(or deformation work)
F?
F?
???F?
? Ⅱ.Derivation:
1,Force on screw dislocation
Criterion:
The work done on crystal by stress must be equal to that
done on dislocation by force,F?
?
A
L
B
F?
b?
?
AA?
BB?
F? Sd L b?
for screw dislocation
)d()()(d SLAbASF ???? ?
microscopic work＝ macroscopic work
bLF ???
2,Force on edge dislocation
x
y
sf
cf
?
F Give,stress field caused
by applies force or other
dislocation
][ ij?
determine,the slip force fs,climb force fc,and total force F on
edge dislocation,
microscopic work＝ macroscopic wok
x
y
sf
cf
?
F
.P.Sb?
?
L?1 )( sfF
Sd
bf
S
A
b
ASf
xy
xy
?
?
??
?????
s
s d1)()(d
if fs make the dislocation move along the contrary direction
bf xys ????
① fs,let dislocation slip a distance dS
② fc,let dislocation climb a distance dS'
SAbASf x ????????? d1)(d
c
cc ?
bf x???? c
③ ftot
2c2st o t fff ??
b?
S?d
?
3,Force on mixed dislocation
given,D a mixed dislocation of unit length withAB OXb //?
),( ?? ??? bAB ?
determine,force on for slip and climbAB
x
y
z
B
A
sf
b?
?
v?
Solution：
① fs:
bf ??s
directs to that part of crystal which contains
extra-half plane.
bL ???
directs to that part of crystal which moves
along,
vl ???
bf xz ??? ?s
b?; For,,For,,
② fc:
x
z
y
o
b?
?
C?
A?
B?
C
A
B
to determine fc,is
resolved into and
BAAB ???
AB
AC CB
CB ?? c o s1 bF xy? AC ?? s in2 bF x??
CAAC ???
)s i nc o s(21 ???? xxyc bFFf ?????
4,General case － Peach-Kochler formula.
l?d
s?d
Ad
F?d
given:
① an mixed dislocation
with arbitrary shape
② ][?
determine,force on segmentl?d
l
Ff
d
d ?? ? nP ?? ???
sFW ?? ddd m ic ?? bPAW ?? ?? )d(d m a c
bSlnbSlPW ???????? ????????? dd)(ddd m a c ?
sl
slbsl
??
?????
dd
dd)(dd
?
????? ?
)dd()( slb ??? ???? ?
slb ??? dd)( ???? ?
－ total stress on plane elementP?
solution:
slbsFWW ????? dd)(dddd m a cm ic ??????? ?
lbF ??? d)(d ???? ?
olbl
Ff ???? ????? )(
d
d ?
,unit vector on dislocation at the point in questionsol?
Example,
][,1 ??AB
)0,s in,c o s( ?? ???ol?
)0,0,(bb ??
x
z
y
b?
?
C?
A?
B?
C
A
B
sf
given:
vbf ??? ??? )(? ]0,0,[bb ?? ]s in,0,[ c o s ?? ??v?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
xz
xy
x
zyzxz
yzyxy
xzxyx
b
b
bb
b
?
?
?
???
???
???
?
0
0
?
??
????
s i n0c os
)(
?
????? yzxyx bbb
kji
vbf
???
???
kbjbib xxyxzxz ??? )s i nc o s()c o s()s i n( ???????? ????
solution:
xzyx bfff ?????? 22s
?
)s inc o s(c ???? xxyz bff ??? ??
xz
xzxz
yx
b
bb
fff
?
????
??
?
??
??
22
s
c o ss in
c o ss in
?
slip:
climb:
§ 5.11 Interaction between dislocation
1,??
2222
22
1
22 ))(1(2
)( b
dx
dxxGbbf
xyx ???
????
???
222
22
1
22 ))(1(2
)3( b
yx
yxyGbbf
xy ???
????
???
Two parallel dislocation of the same (opposite) sign
expel (attract) each other.
d
O
1
2
x
y
Fx
Fy
1b?
2b?
稳 亚稳
12f
xd? d
?45?45
2,$-$
r
bGbbf
zo ?? 2
21
212 ??
x
y
z
O
1b? 2b?
r?
1 2
3,$-⊥ or ⊥ -$
d
O 1
2
x
y
Fx
Fy
1b
?
2b
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
z
b
zyzx
yz
xz
0
0
0
0
0
00
00 2
??
?
?
lb ?? ?? )(?
0
0
0
0
0
2
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
zbzx?
4,-,
⊥ -⊥
$-$
$-⊥ ⊥ -$
x
bbGbf
xyx )1(2
21
22 ??? ?
??????
x
bbGbf
zyx ?? 2
21
22
????????????
022 ????????? xx ff
x
y
z
O
1b
? 2b?
1 2
? ?
21
21
21
222
2
)c o s (
2
)1(2
s i ns i n
bb
x
G
x
bGb
x
bGb
fff xxx
??
????
?
???????
?
??
?
??
??
Two parallel dislocation will attract each other if
<021 bb ?? ?
Two parallel dislocation will expel each other if
> 0
21 bb
?? ?
5,crossed dislocation
Two dislocations perpendicular to each other,but not in
the same plane
x
y
z
O
1b
?
2b
?
C D
A
B
d
given:
dCDABbb,,,,21 ??
determine,12f
solution:
① establish the coordinate aces
② by Peach-Kochler formula
2212 )( olbf ?? ??? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
??
0
0
0
0
00
00
)( 2
2
2
2 zy
zx
yzxz
zy
zx
b
b
b
b ?
?
??
?
?
?
?
)(2
001
0
22
21
22212
yx
kxbGb
kbbb
kji
f zyzyzx
?
??
?
?
?
?
?
?
?
?
?
?
???
?
?
???
?
???
)(2 22
21
12 dx
xbGbff
z ??? ?
x
y
z
O
1b
?
2b
?
C D
A
B
d
Examples and Discussions
Exercise
Thank you !
21
Fundamental of Materials
Prof,Tian Min Bo
Tel,62795426, 62772851
E-mail,tmb@mail.tsinghua.edu.cn
Department of Material Science and Engineering
Tsinghua University,Beijing 100084
Lesson twenty-one
§ 5.10 Forces on dislocation
? Ⅰ.Introduction
1,Definition:
is a fictitious force which is supposed to be responsible for the
motion of dislocation.
∥
Criterion:
The work done on crystal by stress must be equal to that
done on dislocation by force F.
2,Principle of derivation of,
Macroscopic work＝ microscopic work
(or deformation work)
F?
F?
???F?
? Ⅱ.Derivation:
1,Force on screw dislocation
Criterion:
The work done on crystal by stress must be equal to that
done on dislocation by force,F?
?
A
L
B
F?
b?
?
AA?
BB?
F? Sd L b?
for screw dislocation
)d()()(d SLAbASF ???? ?
microscopic work＝ macroscopic work
bLF ???
2,Force on edge dislocation
x
y
sf
cf
?
F Give,stress field caused
by applies force or other
dislocation
][ ij?
determine,the slip force fs,climb force fc,and total force F on
edge dislocation,
microscopic work＝ macroscopic wok
x
y
sf
cf
?
F
.P.Sb?
?
L?1 )( sfF
Sd
bf
S
A
b
ASf
xy
xy
?
?
??
?????
s
s d1)()(d
if fs make the dislocation move along the contrary direction
bf xys ????
① fs,let dislocation slip a distance dS
② fc,let dislocation climb a distance dS'
SAbASf x ????????? d1)(d
c
cc ?
bf x???? c
③ ftot
2c2st o t fff ??
b?
S?d
?
3,Force on mixed dislocation
given,D a mixed dislocation of unit length withAB OXb //?
),( ?? ??? bAB ?
determine,force on for slip and climbAB
x
y
z
B
A
sf
b?
?
v?
Solution：
① fs:
bf ??s
directs to that part of crystal which contains
extra-half plane.
bL ???
directs to that part of crystal which moves
along,
vl ???
bf xz ??? ?s
b?; For,,For,,
② fc:
x
z
y
o
b?
?
C?
A?
B?
C
A
B
to determine fc,is
resolved into and
BAAB ???
AB
AC CB
CB ?? c o s1 bF xy? AC ?? s in2 bF x??
CAAC ???
)s i nc o s(21 ???? xxyc bFFf ?????
4,General case － Peach-Kochler formula.
l?d
s?d
Ad
F?d
given:
① an mixed dislocation
with arbitrary shape
② ][?
determine,force on segmentl?d
l
Ff
d
d ?? ? nP ?? ???
sFW ?? ddd m ic ?? bPAW ?? ?? )d(d m a c
bSlnbSlPW ???????? ????????? dd)(ddd m a c ?
sl
slbsl
??
?????
dd
dd)(dd
?
????? ?
)dd()( slb ??? ???? ?
slb ??? dd)( ???? ?
－ total stress on plane elementP?
solution:
slbsFWW ????? dd)(dddd m a cm ic ??????? ?
lbF ??? d)(d ???? ?
olbl
Ff ???? ????? )(
d
d ?
,unit vector on dislocation at the point in questionsol?
Example,
][,1 ??AB
)0,s in,c o s( ?? ???ol?
)0,0,(bb ??
x
z
y
b?
?
C?
A?
B?
C
A
B
sf
given:
vbf ??? ??? )(? ]0,0,[bb ?? ]s in,0,[ c o s ?? ??v?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
xz
xy
x
zyzxz
yzyxy
xzxyx
b
b
bb
b
?
?
?
???
???
???
?
0
0
?
??
????
s i n0c os
)(
?
????? yzxyx bbb
kji
vbf
???
???
kbjbib xxyxzxz ??? )s i nc o s()c o s()s i n( ???????? ????
solution:
xzyx bfff ?????? 22s
?
)s inc o s(c ???? xxyz bff ??? ??
xz
xzxz
yx
b
bb
fff
?
????
??
?
??
??
22
s
c o ss in
c o ss in
?
slip:
climb:
§ 5.11 Interaction between dislocation
1,??
2222
22
1
22 ))(1(2
)( b
dx
dxxGbbf
xyx ???
????
???
222
22
1
22 ))(1(2
)3( b
yx
yxyGbbf
xy ???
????
???
Two parallel dislocation of the same (opposite) sign
expel (attract) each other.
d
O
1
2
x
y
Fx
Fy
1b?
2b?
稳 亚稳
12f
xd? d
?45?45
2,$-$
r
bGbbf
zo ?? 2
21
212 ??
x
y
z
O
1b? 2b?
r?
1 2
3,$-⊥ or ⊥ -$
d
O 1
2
x
y
Fx
Fy
1b
?
2b
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
z
b
zyzx
yz
xz
0
0
0
0
0
00
00 2
??
?
?
lb ?? ?? )(?
0
0
0
0
0
2
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
zbzx?
4,-,
⊥ -⊥
$-$
$-⊥ ⊥ -$
x
bbGbf
xyx )1(2
21
22 ??? ?
??????
x
bbGbf
zyx ?? 2
21
22
????????????
022 ????????? xx ff
x
y
z
O
1b
? 2b?
1 2
? ?
21
21
21
222
2
)c o s (
2
)1(2
s i ns i n
bb
x
G
x
bGb
x
bGb
fff xxx
??
????
?
???????
?
??
?
??
??
Two parallel dislocation will attract each other if
<021 bb ?? ?
Two parallel dislocation will expel each other if
> 0
21 bb
?? ?
5,crossed dislocation
Two dislocations perpendicular to each other,but not in
the same plane
x
y
z
O
1b
?
2b
?
C D
A
B
d
given:
dCDABbb,,,,21 ??
determine,12f
solution:
① establish the coordinate aces
② by Peach-Kochler formula
2212 )( olbf ?? ??? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
??
0
0
0
0
00
00
)( 2
2
2
2 zy
zx
yzxz
zy
zx
b
b
b
b ?
?
??
?
?
?
?
)(2
001
0
22
21
22212
yx
kxbGb
kbbb
kji
f zyzyzx
?
??
?
?
?
?
?
?
?
?
?
?
???
?
?
???
?
???
)(2 22
21
12 dx
xbGbff
z ??? ?
x
y
z
O
1b
?
2b
?
C D
A
B
d
Examples and Discussions
Exercise
Thank you !
21
