slip lines theory

7/25/2019 Slip lines theory

1/19


2/19

!

"#$%&'%() *%++,

"-../$ 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


3/19

y=3 NH

1

2=2k

> G$/4&2

L() M%44/4

5O

E(*$

"P%/'/$,

B Q


4/19

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 ) "


5/19

=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


6/19

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


7/19

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,


8/19

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


9/19

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

!


10/19

"*(4#($1

,

R B ^

5

Y S

Y 7

>

"

*(4#($1,

K kbZ

22=

12=0

"8 ,

i

12=ksin2=0= =45


11/19

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,


12/19

j B B

!

0 5= !

0=1

F \nC

5 R B

!

0=5.43 !


13/19

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`,


14/19

> -=

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 :


15/19

V

!

5k \%++


16/19

2k=

2k(

!

0)

"h5\%++,

Y B

5j

RS

3=11

! ! * 5 >


17/19

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


18/19

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


19/19

slip lines theory

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