On Stress-Strain State of a Non-Homogeneous Composite in Form of a Package Composed of an Arbitrary Finite Number of Elastic Wedges under Anti-Plain Deformation

5

 

67, №2, 2014

539.3

А Ё

-А А А Ь

А Ь А А

А. ., .А.

: , , - ,

-.

Keywords: composite, elasticity theory, equations of finite-difference, peace-wise homogeneous wedge

Ա. ., .Ա.

:

Gasparyan A.V., Davtyan Z.A.

On Stress-Strain State of a Non-Homogeneous Composite in Form of a Package Composed of an Arbitrary

Finite Number of Elastic Wedges under Anti-Plain Deformation

With the application of Mellin transforms stress-strain state of a piecewise non-homogeneous composite in form of a package composed of an arbitrary finite number of elastic wedges with different physical and geometrical parameters is considered under anti-plain deformation.

ё

-- ,

, ,

а а .

ё

, -

,

.

ё

[1, 2].

- [3].

, ,

.

1. х .

-( , , )

r



z

Oxyz

Ox

6

 

 

1

0

{

,

0

,

}

(

1, )

, (

0, ), 0

2

k k k

i n

z

r

k

n

i

n



     

  



   



     



     

. 1

0

( )

r





_n

( )

r

, ,

,

  

0

  

n,

Oz

r



:

0 0

( , )

( ),

( , )

( ).

z z n

n

r

r

r

r

 _  _





 





 

.

Б а ь, . Т а

 

k

-



_k

:

1

2 2

2 2 2

1

, 1

1

1

0

( ),

( )

0

(

)

k k

k k k

k k k k

k k

z rz k k

w

w

w

r

r

r

r

G

w

G

w

r

r

r

r

r





 

 













 













_{ }

_{ }



_

_



  

 



   





(1.1)

( , )

k k

w



w r



–

а



k

Oz

,



k1

( )

r



k

( )

r

– , ,

1

k

  

  

_k

, (

k



1, )

n

а



k. (1.1)

[4], а а

0( )r   

( )

n r

   G1

n

G

2 G

0  

1    

n

 

y

x

7

0 0





, (1.1) , ё щ

а а а :

1

2

2

1 2

1

0

(

,

1, ),

,

(

1 Re

0; 0

1).

k k

k

k _{k k}

k k

k k

k k

d w

p w

k

n

d

d w

d w

G

G

p

d

d







 









   



 







_{ }

_{ }

_{  }

_

_{  }









(1.2)

(1.2)

1

( , )

_k

cos

_k

sin

(

_{k k}

)

k k

w



w

p

 

A

p

 

B

p



    

_ , (1.3)

k

A

B

_k (1.2)

1 1

1 1

cos

cos

1

(

1, )

sin

sin

sin (

)

k

k k k

k

k k k k k k

A

p

p

k

n

B

pG

p

p

p

 

 





 

























_{  }



_{ }

_

_

_









_

_

.

1 ₁ 1 ₁

1 1

cos

cos

sin

sin

,

(

1, )

sin (

)

sin (

)

k _k k _k k _k k _k

k k

k k k k k k

p

p

p

p

A

B

k

n

pG

p

pG

p

 _  _

 



  





  









  

  

(1.4)

(1.3)

  

_k

ё



k



k1

1

,

(

1,

1)

k k

k k

w

w



k

n











1 1

cos

sin

cos

sin

(

1,

1)

k k k k k k k k

A

p

 

B

p

 

A

_

p

 

B

_

p



k



n



. (1.5)

, (1.4), (1.5)

-k



:

1 _{1 1} 1

1

(

)

0 (

1,

1)

ctg

1

,

,

(

1, ).

sin

k k k

k k k k

k

k k k k k

k k k

a

b

b

a

k

n

p

a

b

k

n

G

p

G

 _{ } 



 



 















    





(1.6)

(1.6)

1 1

1 1

1 1 1

sin(

)

sin(

)

.

sin(

)cos(

)

sin(

)cos(

)

k k

k k k k

k

k k k k k k

G

p

G

p

G

p

p

G

p

p

 

 

  



 

 

 



 





  (1.7)

(1.6) -

1

k k

k k k

a

   



b

f

(1.8)

1

1 k 1 k 1

(

1,

1).

k k k

a

_

 



b

_

 

g

_

k



n



(1.9)

8

1

0,

(

1,

1).

k k

f



g

_



k



n



(1.10)

,

(1.8), (1.9) (1.10).

, (1.4)

k

A

B

_k, (1.3) –

w

k

( , )

p



. ,

:

( , )

( , )

(0

;

1

Re

0; 0

1;

1, ),

c i

p k

k

c i

w r

w

p

r dp

r

c

p

k

n

 



 

 



      



  





.

2. х - х

. - (1.8) (1.9)

, [3]. (1.8)



1

( )



1

( ),

( )

,

( )

(

1,

1)

k k

k k k

k k

P k

Q k

b

a

f

P k

Q k

k

n

b

b



  

 





 





,

0

1 1

1

[1

( )]

(1,

1)

[1

( )]

k k

i

к _i

i j

i r

f

P j

k

n

b

P r

 















 



_



_







_

















(2.1)

(1.9)

1 1 1

1

1 1

1 1

1

1

[1

(

1)]

1

(

1)

(

)

(

1)

(1,

1).

n n n

i

к n

i k

j k i r i

k k

k

g

P r

P j

a

a

b

P k

k

n

a

  





   

 







 

_



 

_



_{ }

_



 

















(2.2)

(2.1) (2.2) (1.10),

(

1,

1)

i

f

i



n



:

1

1 1

1

1 1 1 1

1 1

cos(

)

sin(

)

cos(

)

cos(

)

n

r n

n r i

i i n i n

i k

j j

j k j k

p

G

p

f

p

p



 

 

   



 

 



















9

1

1

1 1

tg(

)

, (

1,

1)

cos(p

)

cos(

)

r

i i k i k

i

j j

j j

G

p

f

k

n

p





 



 



















. (2.3)

1 1

1

1 1

tg(

), (

1, ),

sin(

), (

,

1)

cos(

), (

1, ),

cos(

), (

,

1)

i i i i i i

i n

i r i r

r r i

A

G

p

i

k

B

G

p

i

k n

C

p

i

k

D

p

i

k n

 





 

































(2.3)

1 1

0 1

1

, (

1,

1)

k n

n

i i

k k i i i i

i k i k k k k

C

D

A f

A

f

B

f

k

n

C

D

C

D

 



 





















(2.4)

1

cos(

)

n

n r

r

K

p









,

:

1 1

1 1 1

1 1

cos(

)

;

cos(

)

cos(

)

(

,

1,

1,

1).

n n

n n

i r k r

r i i i j k k

K

K

D

p

D

p

p

C

C

i

k n

k

n

 

  

   



























ё (2.4)

:

, 1

2 1

2 1

1

1

(

1,

1)

, (

1,

1),

(

),

,

,

(

)

k

ki i k

i

n n

k i

n

i i i k k n n i

i

ki i n i

k k

L f

h

k

n

C

A

A C

A

i

k

h

C

X

X

f

L

C

K

C

A C

i

k



  











_

_

_

_{  }

_



_



 



_







[1].

3. .

, ,



₀

( )

r

2

( )

r



,

  

₀

  

₂

. ,

- (1.7)

0 2

2 1

1

2 1 1 2

sin(

)

sin(

)

sin(

)cos(

) sin(

)cos(

)

p

p

p

p

p

p



  

 

 





 





(3.1)

2 2 1

,

1 1 0

,

G G

2 1

10

1 1 1

1

( )

2

c i p

c i

r

r

dp

i

   

 











,

[5, 6] ё

1 0 2

0 0

1

( )

( , ) ( )

( , ) ( )

2

r

K x r

x dx

L x r

x dx

r

 









_









_

 







(3.2)

2

1 1 2

0

1

2 1 1 2

0

sh(

) cos( ln )

( , )

sh(

)ch(

) sh(

)ch(

)

sh(

) cos( ln )

( , )

( 0

)

sh(

)ch(

) sh(

)ch(

)

x

s

s

ds

r

K x r

s

s

s

s

x

s

s

ds

r

L x r

r

s

s

s

s















 









  





 









а ,

, . .

0

( )

r

2

( )

r

T

(

r

r

0

)



 

  

,

( )

r



– .





1 0 0

0 2

0

1 1 2

0

0 1

0

2 1 1 2

0

( )

( , )

( , )

4

sh(

) cos( ln

)

( , )

sh(

)ch(

) sh(

)ch(

)

sh(

) cos( ln

)

( , )

( 0

)

sh(

)ch(

) sh(

)ch(

)

T

r

K r r

L r r

r

r

s

s

ds

r

K r r

s

s

s

s

r

s

s

ds

r

L r r

r

s

s

s

s

























 









  





 









(3.3)

(0)

0 1 1

,

,

( )

a

(

)

r

а

r

а

a

T

 

      

(3.3)





(0) 1

1

( )

( , )

( , )

4

K

L

  

     



. (3.4)

) . 2-4

 

1(0)

( )





1

,

2

2

4





11

Ф . 2

Ф . 3 . 4

,

 

1

,

 

1

,

(

 

1

).

)

r



r

₀

(

  

)

(3.3) (3.4)





(0) 1

2

1 1 2

0

1

2 1 1 2

0

1

( )

( , )

( , )

4

sh(

)

( , )

sh(

)ch(

) sh(

)ch(

)

sh(

)

( , )

( 0

)

sh(

)ch(

) sh(

)ch(

)

K

L

s

ds

K

s

s

s

s

s

ds

L

s

s

s

s



 



 

     





  





 







  

   





 









(3.5)

2 4 6 8 10 x

0.02 0.04 0.06

2 4 6 8 10 x 0.005

0.010 0.015 0.020 0.025 0.030 0.035

t10x

2 4 6 8 10 x 0.005

0.010 0.015 0.020 0.025

t10x

1

 

 

10

 

 

 

1

 

 

0.1

 

 

 

10

 

0.1

 

 

1

 

 

10

 

 

2

 

 

7

 

 

5

12

. 5 . 6

. 5

 

₁(0)

( )

0,1; 0,5;1;3; 7

 

1

,

2

3

4





 

 

.

, ё



.

) На . 6

 

1(0)

( )

 

1



1

2

; ; ; ;

6 4 3 2

3

    

 

₂

3



 

.

3.2 ё -

    

_{1 2} (1.7)





0 2

1

1

cos(

p

)

  

 

 



. (3.6)

Е

  

0 2 ,

0 1

cos(

p

)



 



, . .

ё



, . 4.

, (3.6) ,

[5, 6]

 

 

 

 

2 2

1 0 2

0 0

1

( )

( )

( )

2

(

1)

₁

₁

x r

x r

r

x

dx

x

dx

r

_{x r}

_{x r}

 

   

 

 













 

 





  

_

_









. (3.7)

I. ,

ё ,

0

( )

r

2

( )

r

P



 



, (3.7)

1

( )

, 0

2 cos

2

P

r







  



.

2 4 6 8 10 h

0.02 0.04 0.06 0.08 0.10 0.12 0.14

t10h

2 4 6 8 10 m 0.05

0.10 0.15 0.20

t101

1

4

  

1

2

3

  

1

2

  

1

3

  

7

 

0,1

 

0, 5

13









2 0 1

0

( )

4

₁

r r

Т

r

r

_{r r}

 

 





 

.

1. . ., . . ё

ё

// . . . 2010. .63. № 2. .10-20.

2. . ., . .

// . - .

28 – 1 2009, , , .186-189.

3. . . . : , 1967. 375 .

4. . . . :

, 1967. 403 .

5. ., . . .1. .: ,

1969. 343 .

6. . ., . . , , .

.: , 1971. 1108 .

х: А

. .- . ., ,

А : 0019, , . 24/2

.: (+37410) 52-48-90, -mail:anush@mechins.sci.am

А

. .- . ., ,

А : 0019, , . 24/2

.: (+37410) 52-48-90, -mail: anush@mechins.sci.am