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7/25/2019 Slip lines theory
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7/25/2019 Slip lines theory
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!
"#$%&'%() *%++,
"-../$ 0(-)1 2)2+34%4,5
!
56
!
5
7,8
9,: ;
!
"?3, 5=
@ A
B C/D3 E%44/4
d z=0=d
[z12 ( +y) ]=0=z=12 (+y)
F B
5
> G$/4&2
1
2=y=2k
HIJ
K L() M%44/4
1
2=
2
3y
7/25/2019 Slip lines theory
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y=3 NH
1
2=2k
> G$/4&2
L() M%44/4
5O
E(*$
"P%/'/$,
B Q
7/25/2019 Slip lines theory
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RS T U
x=1
2(1+2)+
1
2( 12 )cos 2 (1,x ) "7,
y=1
2 ( 1+2 )1
2( 12 )cos2 (1,x ) "9,
xy=1
2(12 )sin2(1,x ) "
7/25/2019 Slip lines theory
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=ksin2 (4)
y=+ksin2 (5)
xy=kcos2 (6)
= B
d
d x1
2k(cos2 d d x1
+sin2d
d x2)=0 "W,
d
d x2
2k
(sin 2
d
d x1
co s 2d
d x2
)=0 "X,
Y B B W!X
B 5 T
B
: ; Z
d x2
d x1=
1
a(b b2ac) b2ac>0 "[,
\ + B
B [
O B "W, "X,
x'1 , x '2 5U B
7/25/2019 Slip lines theory
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T B W!X
d 2
d x'
1
2k d
d x'
1
=0 "[,
d 2
d x '2+2k
d
d x '2=0 "7],
O B [ 7]
22k =C
1=constant along a line "77,
2+2k =C
2=constant alongb line
(12)
=
2
77 79
B !e n cky
+2k=C1(aline ) "7
7/25/2019 Slip lines theory
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7,Y S
9,
"Y V
\/)&H3,
T _ P
B \/)&H3Z"#
2"=2#+2k , "7`,
T a b
2C=2"2k(c") (16)
>S B 7` 7c
2C2#=2k(2"#C) (17)
= ! =db
Y #"=$C "7X,
7/25/2019 Slip lines theory
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R S % =!
S# B \/)&H3
C2
5
C2
S 5T
5 RS
C2
&'( B \/)&H3 )
5O C
1
^,=
!! C
1,C
2
e
`,: V \/)&H3ZF
S
f
d*++d =0alongaline
g fZ1
*=
d
d s( "7[,
d+*d=0along b line h fZ
1
+=
d
d sa "9],
V
B
5 F E5C/D3
i B 77 79
,=C1
&C2 B W X
7/25/2019 Slip lines theory
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d ,
d x+
d,
dytan=0
(21)
d&
dx
d&
dy
cot=0(22)
j B 5 V
;
V =( , & ) y=y ( , & )
O f
- ( , & )=" ( , & )
" ( , y )
=d
dx
d &
dy
d
dy
d&
dx
.0
id
dx=-
dy
d& !/,
/ y=-
dx
/& !d &
dx=-
dy
d !d&
dy=-
dx
d
O B S
dy
d,+
dx
d,cot =0
(23)
dy
d&
dx
d &tan=0 (24 )
R
!
7/25/2019 Slip lines theory
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"*(4#($1
,
R B ^
5
Y S
Y 7
>
"
*(4#($1,
K kbZ
22=
12=0
"8 ,
i
12=ksin2=0= =45
7/25/2019 Slip lines theory
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22=+sin2k=0==k
11=ksin2=2k
j "OO, l
i
12=ksin2=0= k=45
6 8B \/)&H3 S
2k =2 k ==+2k( )= =(1+)k
22=+2ksin (2)=(2+)k
me9H" (1+)=
2k=2.57
V
j (
5i
"\(4#($1,
7/25/2019 Slip lines theory
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j B B
!
0 5= !
0=1
F \nC
5 R B
!
0=5.43 !
7/25/2019 Slip lines theory
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R
B \/)&H3 !
5
y= , =+k m
T "]!7,
B \/)&H3(0,1)=+2k - 5T
"7!7,
(1,1)=(0,1)2k - (=+2k(-(-)=+k+2k(--() "9`,
7/25/2019 Slip lines theory
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> -=
12 -(=
12 5>
")!), -=
n12
-(=n12
i 9`5 (n ,n)=+2k(1+2-)
j B
5 i
+2k(1+- )
dy=0=0
1 /2
[]
2x=0
1 /2
13e]eJ
1
2=2k( 12 )+2k
0
1/2
2- dy==2k+4k
1
0
1/2
2- dy
j S 5j :
7/25/2019 Slip lines theory
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V
!
5k \%++
7/25/2019 Slip lines theory
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2k=
2k(
!
0)
"h5\%++,
Y B
5j
RS
3=11
! ! * 5 >
7/25/2019 Slip lines theory
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5
=
$o
2sina
1+2 sina 5>
5
V
5
TS ?!3
x=(x '1 )cosy 'sin
y=(x '1 ) sin+y 'cos
y= , =+k 5Y
"]!7,
(0,1)=2 k
12=+k2k
12
7/25/2019 Slip lines theory
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j ")!)Q7, !
p \/)&H3 S -=n
12
-(=n 12 5
i ( n, n+1)=+2k+2k(2n+1 )
12=+2k[1+
(2n+1 ) 12
]
> B B ! *e 1 4
i 25=0
14
x dy=0
14
(+2k[1+
(2n+1 ) 12 ])dy=0
2k=1+
12140
14
(2n+1)d y
j S !
1 4 5
j Ze=
2sina
14+2 sina 5
!"!
h5\%++Z G*/ E2'*/M2'%&2+ G*/($3 (# m+24'%&%'3
b211/+I\(4#($1Z E/'2+ q($M%)r E/&*2)%&4 2)1 E/'2++-$r3
C5E l2&*2)(DZ q-)12M/)'2+4 (# '*/ G*/($3 (# m+24'%&%'3
P%/'/$Z E/&*2)%&2+ E/'2++-$r3
s5s g(H(+(D4H3 ZG*/($3 (# .+24'%&%'3
7/25/2019 Slip lines theory
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