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Antiplane contact problem for elastic half-space with cracks and stringers

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А А Ь А А А А

69, №1, 2016

593.3

А А А А А А А

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ы : , , , ,

Keywords: contact problem, stringer, crack, half-space, singular equation

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. ., . .

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– :

:

, :

Aghayan K.L, Zakaryan V.G

Antiplane contact problem for elastic half-space with cracks and stringers

The contact problem of load transfer of a thin elastic plate to the elastic half-space is considered. The half-space weakened by a system of finite length collinear tunnel cracks which are perpendicular to the half-space boundary. The border of half-space reinforced by stringers. By external forces applied to the plates and the banks of the cracks, the half-space - plate system deformed under antiplane strain. In the frame of well-known Melan model for stringer the solution of the problem is reduced to a system of singular integral equations, whose solution is built by numerically - analytical method of mechanical quadratures. The behavior of the contact stresses and stress intensity factors at the crack tip and the end points of stringers is investigated.

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,

Oxyz

.

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;

1 2

;

(1)

0;

1, 2,...,

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x

a l

j j

y

l

j

z

l

j

j

n

 

 

 

-,

y

0

,

 

k

c

k

 

x

b y

k

;

0;

z

 

1

(

a

k

c

k

b

k

a

k

)

, ( )

 k s

h

G

s k

,

k

1, 2,...,

n

1

.

. 1

, : )

0

xz

a

j

w a

j

0

; )

ё

yz

x

,

h

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, )

x

c

k

x

b

k

T

c( )k

T

b( )k ( . 1). ,

Oz z

 

.

– ,

G

s( )k

G

,

( )k

s k k

h

 

b

c

. Э

[3-4]. , ,

,

s

w x y

, . .

w x y

s

,

w x

s

 

.

, ,

.

,

Oxy

1

b 1

c c2 b2 x

y z

 1 1 l

 2 1 l

 2 2 l

 1 2 l 1

a a2

1 n

cbn1

 1 1 n l

 2 1 n l

1 n a

 1 n l

 2 n l

(3)

0

y

,

1,

j

x

a

j

n

y

0

.

0

y

:

2 2

2 2

0

w

w

x

y

,

   

x

,

y

0

(1.1)

 

 

 

0

,

,

1,

0,

,

j

j j

x a

j j j

w x y

G

y

y

L

x

j

n

w a

y

w

y

y

L

 

 

 





(1.2)

 

 

1

0

1

,

( , )

0,

(

)

k k

n

k y

k

x

x

w x y

x

y

x



 

 

(

k

1,

n

1)

(1.3)

(1) (2)

,

,

0,

j j j j k k k

L

l

 

y

l

x

a

 

y

c

 

x

b

[3-5]

 

 

 

( )

( ) ( ) ( ) ( )

0

( , )

1

k

x k

k c

s k

k k k k

y s s c s s

T

w x y

q

s

s ds

x

h G

h G

 

(1.4)

,

1,

1

k

x



k

n

.

,

w x y

– , ,

,

j

 

y

j

- ,

w

j

 

y

j

-,

k

 

x

– ,

,

q

s( )k

 

x

– , . ,

(1.2).

(1.1) – (1.3),

 

x

, . . ё

,

. ,

. (1.1) – (1.3)

.

(1.1), (1.2),

w

,

y

(4)

2

2 2

1

,

,

( )

( )

k

,

0

n

i a

k k

k

d w

y

w

y

i g y

f

y e

y

y

 

 

,

 

,

i x

w

y

w x y e dx



, (1.5)

 

0,

0,

k k k

g

y

w a

y

w a

y

(1.6)

 

1

 

 

k k k

f

y

y

y

G

 

 

(1.7)

k

- .

(1.3) :

 

0

,

1

y

dw

y

dy

G

 

(1.8)

,

f

k

 

y

,

g

k

 

y

 

x

, (1.5),

(1.8)

y

 

,

1 0

1

,

(

)

sgn

( )

( )

2

2

( ),

0

k

n

y y i a

k k

k

y

i

w

y

e

e

g

f

e

d

e

y

G

      

 

 

 

 



(1.9)

, :

 

 

0

k k

g

y

f

y

,

y

L

k,

k

1,

n

(1.10)

, (1.9) (1.1) –

(1.3)

 

 

2

 

2

 

2

2

 

1

1

,

2

k

n

k k

k

k L k k

x a

x a

w x y

g

d

x

a

y

x a

y

  

 

  

  

 

2

2

 

2

2

 

1

1

1

1

ln

ln

2

k

n

k

k L k k

f

d

x

a

y

x a

y

  

 

  

  

 

2 2 2

1

ln

const

2

y

t dt

G

t

x

y



(1.11)

,

g

j

 

y

f

j

 

y

, (1.11),

, ё , ,

. , ,

, . . , ,

(1.2),

n

2

n

.

, , , (1.2)

y

,

(5)

 

 

2 2

0

1

1

1

1

1

( )

2

2

j j

j

j j

x a L j

a

t

w

g

dt

f

y

t dt

x

y

y

G

a

t

y

  





 

1

1

(1

)

( , )

( )

( , )

( )

,

2

k

n

ik jk k jk k j

k L

K

y g

R

y f

d

y

L

 

 

 

(1.12)

 

 

0

1

1

1

1

2

2

j j

j j

L x a

w

g

y

f

d

y

 

y

y

 

  

 

 

( ) 1

1

(1

)

( , )

( )

2

k

n

jk jk k

k L

R

y g

d

 

  

( )

2 2

1

( )

( )

,

jk k j

j

y

d

K

f

d

y

L

G

a

y

 



  



 

(1.13)

2

2

2

2

,

jk

j k j k

y

y

K

y

y

a

a

y

a

a

 

 

 

 

(1.14)

2

2

2

2

,

j k j k

jk

j k j k

a

a

a

a

R

y

y

a

a

y

a

a





(1.15)

 

k k

g

t

dg

dt

,

ij – .

, (1.12) (1.13) ,

1 / (

 

y

)

.

, (1.12), (1.13), (1.2),

, , ,

2

n

. ,

ё ё

, , ,

-, .

, (1.11) (1.12) (1.13)

, ё

. , :

ё ; ё ,

; ё .

2. .

, .

(1.4) .

x

, , (1.11), :

2 2

1 0

( , )

1

( )

(

)

k

n

k k

y L k

w x y

g

x

 

a

x

 

 

(6)

2 2

( )

1

( )

,

(

)

j j k k k

t

a

x

f

d

dt

a

x

G

t

x

 

 

(2.1)

( )

j

x

– ,

(

1, 2,...,

1)

j

j

n

.

(2.1) (1.4) , ,

. , 1,

, .

,

(

n

2),

(1.2) ( . 1), (1.12) (2.1)

( ), (

1, 2)

k

g t

k

1

( )

x

.

1 1 2

1 1

1 12 2

2 2 1

( )

1

1

1

1

1

( )

( , )

( )

(

)

2

L L

a

t

t

dt

g t dt

K

t y g t dt

a

t

y

G

t

y

t

y

 

2 1 1 12 2

( )

( )

1

1

( , ) ( ) ,

2

L

y

y

R

t y f t dt

G

  

 

(1) (2) 1

(

1 1

)

y

 

l

 

y

l

(2.2)

 

1 1 2

2 1

21 1 2 2

2

( )

1

1

1

1

1

( , ) ( )

(

)

2

L

2

L

a

t

t

dt

K

t y g t dt

g t dt

a

t

y

G

t

y

t

y

 

 

 

   

1 2 2 21 1

1

1

,

2

2

L

y

y

R

t y f t dt

G

  

 

(1) (2) 2

(

2 2

)

y

 

l

 

y

l

(2.3)

  

 

 

1

2 2

1 2 2 2 2

1 1

1

k k

k k

k

k L k k L k

t

t g

t dt

x

a

x t

dt

f

t dt

t

x

G

t

a

x

x a

t

  

 

 

   1

1

1

1 1 1

c s

s s

T

x t q

t dt

h G

  

x

 

1

c

1

 

x

b

1

(2.4)

   1 1 1

G h G

s s

  

,

q

 s1

 

t

– ,

,

f

k

 

t

– , (1.7),

K

ij

 

t y

,

,

 

,

ij

R

t y

(1.14), (1.15).

(2.2)-(2.4) ё

 

 

 

1 1

1 1

1

s c b

s ds

q

s ds T

T

 

,

 

0

j

j L

g

s ds

. (2.5)

,

– . ,

, . .

l

k 1

0

,

(7)

3. ы ,

,

( . 2). ,

 1

0

j

l

,

l

 j2

l

,

j

1, 2

,

a

2

  

a

a

1,

c

1

 

c

,

b

c

,

0,

 

 

xz

a

y

a

y

P y

 

  

 

,

  

xz

a

0,

y

 

a

 

y

Q y

 

,

0,

 

 

xz

a

y

a

y

Q y

 

 

, xz

a

0,

y

a

 

y

P y

 

 

 

,

 1

 

 

0

s

q

x

q

x

,

q

0

 

x

q

0

 

x

,

T

c

 

T

c

T

0,

Oy

, ,

g y

 

g

1

 

y

 

g

2

 

y

.

.2

, (2.2) - (2.4),

 

1

x

g y

 

:

  

 

1 2 2 2 2

0

1

c l

c

x s

s ds

g

d

s x

a

x

a x

  

 

  

  

2

2

 

  

0    1 1

2 2

0

l c

a

c s s

T

a

x

a

x

R

d

x s q

s ds

h G

a

x

a x

    

  

  

  

c

x

c

(3.1)

  

2 1 2 2

2 2

2

 

0

1

1

4

4

c l

c

a t

t

t

y

t

y

dt

g t dt

t

y

t

y

a t

y

a

t

y

a

t

y

 

 

 

2

2

 

 

2 2

0

4

4

2

l

a

a

R

t dt

R

y

a

t

y

a

t

y

 

 

 

0

 

y

l

(3.2)

 

1

 

 

R

y

P y

Q y

G

;          1 1

s s

G

h G

  

 

(2.5)

a



a

 a

  a

 

a

a

c c

l O

y

x z

 

0

q x

c

T

c

(8)

 

0

 

2

0

c c

c c

s ds

q

x dx

T

 

. (3.3)

(3.1),

(3.2) (3.3), .

,

. .2 ,

: 1)

c

a

– ; 2)

c

 

a

0

– ; 3)

c

 

a

0

,

, ,

. .

[12,13],

g y

 

y

l

0

y

.

, ,

 

x

g y

 

. ,

   

c

c

Gg

 

0

   

,

y

l

-

g y

 

.

(3.1), (3.2) (3.3), ё

( 1,1),

[14].

Ч ,

c

a

c

 

a

0

, ,

,

ё , . .

P t

( )

Q t

( )

0;

T

c

0,

0

/

1,

q

G

 

0.5

.

.3 .4

c

a

( .3, 4),

0

c

 

a

– ( .5, 6). .3 5

, . 4 6 –

2 /

c l

 

, . .

. .1

III

K

. А

,

-1.0 -0.5 0.5 1.0

0.05 0.10 0.15 0.20

G

 4



2



2 3

 0.2

-1.0 -0.5 0.5 1.0

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1 g

0.2

 2 3

 2

(9)

.5 .6

1.

K

III

4 2 0.67 0.2

c

a

-0.0974 -0.0796 -0.0504 -0.0285

c

a

-0.1504 -0.1241 -0.102 -0.0883

.

,

, ,

.

,

.

.

.

А А

1. Greif R., Sanders J.L. (Jr.) The effect of a stringer on the stress in a cracked sheet. //Trans. ASME, Ser. E: J. appl. Mech., 32 (1965), pp. 59–66.

2. А . . ,

. // .: «

». : : А А , 1985, .26 – 32.

3. Э. .

. // . . . :

. , 1986, № 5, .130 – 140.

4. . . ,

. // .: «

ё ». : . А А . 1993, .129 – 143.

5. А . ., . . ,

.// .: «

ё ». : . А А . 1993, .42 – 47.

6. . .

. // . . .:

«А ». 2012, .74 – 78.

-1.0 -0.5 0.5 1.0

0.02 0.04 0.06 0.08 0.10

G

 4



2



2 3

 0.2 -1.0 -0.5 0.5 1.0

-0.20

-0.15

-0.10

-0.05 g

4   2

 

2 3   0.2

(10)

7. А . . . // . «

ё ». 2012, .42-47. 8. А . ., А. . ё

- . // .

. .: «А », – 2015,

.38 – 42.

9. А . ., . ., А . . А ё

. // А

А . . 2009. .62. №4. .16 – 22.

10. . .

. // . . .: «А », – 2015, .204 – 208.

11. . .

. //

А А . . 2014. .67. № 3. .3 – 16.

12. А . . .

// . 1968. .32. . 4. .632 – 646. 13. . 2. .: , 1975. 763 . 14. А. .

. // . А

А . . 2000. .53. № 3. .12-19.

:

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