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15.1.
−
−
−
+
−
−=+
−
−=+
−
−=
−
=
log
)1(8
1
)1(8
log
)1(8
)43(
case)strain plane(for )()()()(2
log
)1(4
)(,log
)1(4
)(
potentials thechoose methods, riablecomplex va Using
____________
z
ib
z
bi
zz
ib
ivu
zzzzivu
z
bi
zz
bi
z
15.1. Continued
+
−
+
+
−
−
−=
+
+
−
+
−
−
=
+
−
−
+
+
−
=
222
22
222
22
222
22
222
22
222
22
222
22
)(
)(
)(
)(
)1(2
)(
)3(
)(
)(
)1(2
)(
)(
)(
)3(
)1(2
scoordinateCartesian of in terms stresses theExpressing
yx
yxy
b
yx
yxx
b
yx
yxx
b
yx
xyy
b
yx
yxx
b
yx
yxy
b
yxxy
yxy
yxx
15.2.
+
−
−
+
+
−
=
+
−+
−
+
+
−
=
=
==
=
+−
−+
−
−
−=
+−
+
=
−
−
)(
)(
)1(2
1
2)(
2)(
)1(2
1
2
: FieldStrain
0][,tan
2
][ : BehaviorCyclic
)1(2
1
)log(
)1(4
21
2
,
)1(2
1
tan
2
:on DislocatiEdge
222
22
22222
222
22
1
22
2
22
22
1
yx
xyy
yx
yb
yx
yxyyx
yx
yb
x
u
e
vb
x
yb
u
yx
y
yx
b
v
yx
xy
x
yb
u
x
C
C
C
15.3.
0cos
sin
sin
cos
cossin
2
sin
cos
2
sincos
0)sin(coscossincossin
0
0cossin2cossin
0cossin2sincos
:sCoordinate lCylindrica In
2
2,
2
2
0
: FieldStress
42
1
,
42
1
,0
: FieldStrain
tan
2
][ : BehaviorCyclic
tan
2
,0
:ocationScrew Disl
22
22
22
22
22
2222
2222
1
1
=
−
=+=
=
+
=−=
=−++−=
=
=−+=
=++=
+
−==
+
==
====
+
−=
=
+
=
=====
=
=
===
−
−
r
rb
r
r
r
yx
yb
e
yx
xb
e
yx
yb
x
w
e
yx
xb
y
w
eeeee
b
x
yb
w
x
yb
wvu
zxyzzr
zxyzz
xyyxr
z
xyyx
xyyxr
xzxzyzyz
xyzyx
xzyzxyzyx
C
C
15.4.
02cos
)cossin(
)cossin(
02sin
)sincos(
, radius ofCylinder on ForcesSum
cos,sin
:on Dislocatifor Edge Solution Stress
2
0
2
0
22
2
0
2
0
2
0
2
0
==
+−=
+=
=−=
−=
=−==
dbB
dbB
daaF
dbB
daaF
a
r
bB
r
bB
rr
y
rr
x
rr
15.5.
22
by given ismoment twistingresultant theand
,not vanish will length, finite ofcylinder a of ends theOn
2
,0
:ocationScrew Dislfor FieldStress
2
00
2
00
2
0
ba
drrbdrrd
b
drrrdT
r
b
aaa
z
z
zrzrzr
==
==
======
15.6.
o
R
R
R
R
screw
co
zz
R
b
dr
r
b
rdrdUW
RrR
r
b
U
o
c
o
c
4
2
2
1
: Regionin ocationScrew Dislfor Energy Strain
42
1
2
1
:ocationScrew Dislfor nsity Energy DeStrain
2
2
2
2
0
22
2
==
==
x
y
rad
rad
a
15.7.
c
o
c
o
R
R
R
R
screw
co
rrrr
rr
r
rrrr
rr
R
R
b
R
R
Bb
drd
Bb
rdrdUW
RrR
r
Bb
r
Bb
r
Bb
r
Bb
r
Bb
eeU
r
bB
r
bB
EE
e
r
bB
EE
e
r
bB
r
bB
o
o
log
)1(4
log)1(2
2
)sin21(
1
: Regionin on Dislocatifor Edge Energy Strain
)sin21(
2
sin)cos(sin
2
cos
2
sin
2
21
)(
2
1
:on Dislocatifor Edgensity Energy DeStrain
cos
1
sin
2
21
2
21
)21(
1
])1[(
1
sin
2
21
2
21
)21(
1
])1[(
1
:Strain) (Plane on Dislocatifor Edge FieldStrain
cos,sin:on Dislocatifor Edge FieldStress
222
2
0
2
22
2
0
2
2
22
2
2
22
22
2
22
2
2
22
2
2
22
−
=−
=
−
==
−
=
−+
=
+
−
=++=
=
=
−
−=
−
=−
+
=−−
+
=
−
−=
−
=−
+
=−−
+
=
=−==
15.8.
sin
2
)43(
sinsincoscoscos
cos)1(2
cossinsincossin
(B.8), relations tiontransformant vector displaceme theUsing
)21(
3
,)21(
3
,
33
)21(
3
,)21(
3
,)21(
3
))(21(
3
)43(
2
,
2
,
22
)43(
2
,0,0,0:]1,0,0[ case specail For the
))(21(
3
3
,,,
))(21(2
)43(
2
)43()1(4)()1(4)(2
)-(18
1
where,,0 :s Unit Load withState Kelvin
2
2
32
2
3523
2
3
32
2
32
2
3
33
2
2
332
3
2
i
23
35
,
3
,
3
,
33
,
,,,,
,
2
3
,
i
,
k
i
−
−=−+=
−
=++=
−+−=
−+−=−=−=
−−−=
−−−=
−−−=
−
+
−+−=
−+
=
=
=
=
−+
=
−=
=
=
−
+
−+−=
+
−=
=
==
=
++−−
−
=
−+
=
−+=
−+−=
−−
+
=
=
−=
=
z
z
z
z
z
y
z
x
z
z
z
y
z
xR
z
zx
z
yz
z
xy
z
z
z
y
z
x
ij
j
i
ji
z
z
z
z
y
z
x
i
i
z
i
zz
ij
i
jα
j
iα
ji
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ijkα
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kα
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jα
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R
C
uuuu
R
C
uuuu
x
R
xz
R
C
y
R
yz
R
C
R
Czxy
R
zxy
R
C
z
R
z
R
C
z
R
zy
R
C
z
R
zx
R
C
zxx
xzx
C
R
z
R
C
u
R
Czy
u
R
Czx
R
zx
R
C
u
R
zx
R
C
u
R
C
a
xxx
R
xxx
R
C
R
C
xx
R
C
R
x
C
R
x
C
R
x
C
R
x
C
x
R
xx
R
C
u
RR
xx
C
R
C
R
x
Cxu
C
R
C
15.8. Continued
)sin(cossincossinsincossinsin
0
sinsincossin
)sin(coscoscossincoscossincos
sin
)21(
cos)cos(sin
sin)cos(sincossincossin2
cossinsincossincoscossin
cos
)21(cossin2cossin
cos
)21(
coscossin2sincossin2cossincos2
sinsincoscoscos
cos
)2(2
coscossin2sincossin2cossinsin2
cossinsincossin
(B.9), relations ation transformstress theUsing
22
22
22
22
22
22
2
22222
2
22222
−++−=
=
+−
−++−=
−=
−−
−−+
−+=
−=−+=
−=
−−+
++=
−−=
+++
++=
z
xy
z
y
z
xR
z
zx
z
yz
z
xy
z
y
z
x
2
z
zx
z
yz
z
xy
z
z
z
y
z
xR
2
z
xy
z
y
z
x
2
z
zx
z
yz
z
xy
z
z
z
y
z
x
2
z
zx
z
yz
z
xy
z
z
z
y
z
xR
R
C
R
C
R
C
R
C
15.9.
( )
R
R
R
uuu
uuuu
R
uuuu
R
xx
RRR
xx
ee
RR
xx
uue
R
x
e
R
x
u
R
x
R
x
R
x
R
x
RR
xu
R
x
RR
x
R
xx
R
R
x
=
R
=
o
xy
o
y
o
x
o
zx
yz
xy
o
z
o
y
o
x
o
o
zx
o
yz
o
xy
z
y
x
R
y
x
o
z
o
y
o
x
o
o
z
o
y
o
x
o
R
ij
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o
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o
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o
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k
k
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o
i
i
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o
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k
o
o
i
i
i
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o
i
o
3
22
3
22222
2
2
2335
35
,,
,
3
3
333
3
,
i
,
k
3
,
,
2
3
1
cossin2cossin
1
coscossin2sincossin2cossincos2
sinsincoscoscos
2
coscossin2sincossin2cossinsin2
cossinsincossin
(B.9) relations mation transforuse ,components stress sphericalFor
0cossin
0sinsincoscoscos
2
1
cossinsincossin
(B.8) relations ation transformuse ,componentsnt displaceme sphericalFor
3
1
3
2
3
2
1
2
1
,0
2
1
2
)21(
)1(2
)21(
1
)21(
)1(21
)21(2
11
)21(2
3
)1(4)(2
1
,, that Noting
1
)21(2
,
1
)21(2
3
:nCompressio ofCenter
−=−+=
−=
−−+
++=
+++
++=
=+−=
=−+=
−=++=
−=
−=+=
+−
−=+==
−=
−=
−=
−
−
−
−
=
−
−
−
−
+
−
−=
−−
+
=
−=
==
−
−
−
15.9. Continued
0
sincoscoscos
)sin(cossincossinsincossinsin
0
sinsincossin
)sin(coscoscossincoscossincos
0
cos)cos(sin
sin)cos(sincossincossin2
cossinsincossincoscossin
22
22
22
22
22
=
−+
−++−=
=
+−
−++−=
=
−−
−−+
−+=
o
zx
o
yz
o
xy
o
y
o
x
o
R
o
zx
o
yz
o
xy
o
y
o
x
o
o
zx
o
yz
o
xy
o
z
o
y
o
x
o
R
15.10.
−
−
+
=
++−
−
−=
++−
=
−
−
+
=
+
−
−
+
=
++−
−
−=
++−
=
++=++−=
−
=
++−
−
=
++−
−
=
−=−=
ax
R
x
xx
x
xxx
xd
x
xxx
d
x
u
R
ax
R
x
xx
x
xx
xax
R
xx
xx
R
xxx
xdx
xxx
d
x
u
xxxRxxaxR
R
R
xxx
xdx
d
xxx
x
u
xxaxd
ax
aa
ox
aa
ox
aa
ox
oo
aoox
ˆ
)(
2
1
])[(
)(
2
])[(
2
ˆ
)(
2
1
)(
ˆ
11
2
1
])[(
)(
2
])[(
2
and)(
ˆ
where,
1
ˆ
1
2
1
])[(
)(
2
])[(
)(
2
1
),,();(where,);()(
by,given is to0 from axis- along Dilatation of Centers of line afor state The
11
3
2
3
02/33
3
2
2
2
1
1
3
02/33
3
2
2
2
1
3
3
11
3
3
2
2
2
3
3
2
2
21
3
3
2
2
21
02/33
3
2
2
2
1
12
02/33
3
2
2
2
1
2
2
3
3
2
2
2
1
3
3
2
2
2
1
02/33
3
2
2
2
1
2
11
02/33
3
2
2
2
1
1
1
321
0
1
SSSS xxx
