Hybrid and Optimum Designing Composit Plates

Ա Ա Ա Ի Ի Ո Թ Ո ԻԱ Ա Ի Ա Ա ԻԱ Ի Ա Ի

60, 1_{1, 2007}

539.3

. .

: ,

, , ,

Keywords: the hybrid and optimum designing, the reinforced designs, equal-deformability, equal crack resistance of coverings, designs of the minimal weight.

. .

,

:

: ,

:

Yu.V. Nemirovskii

Hybrid and Optimum Designing Composit Plates

The problem of hybrid designing of composit designs of layer-fibrous structure is formulated consists in de-termination of a site of phase materials and a geometrical roll forming of layers for maintenance of necessary operational and mass-dimensional qualities of a product. Illustrative instances of hybrid designing of ring plates are observed at a loading in a plane and at a cross-bending. It is shown, that designs of the minimal weight make a subclass of hybrid designs.

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В

[3].

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:

( )

_{( )}

_{( )}

2

_{( )}

_{( )}

1 2

2

0,

2

d rN

r

N

r

r

r

h r

dr

−

+ ω ϕ

=

ϕ

=

g

⎡

⎣

δρ

+ ρ

⎤

⎦

(1)

1 11 1 12 2

,

2 21 1 22 2

N

=

a

ε +

a

ε

N

=

a

ε +

a

ε

(2)

( )

2 2 2

1

,

2

,

1

,

3 1 13 2 23

d r

du

u

l

l

dr

r

dr

ε

ε =

ε =

= ε ε = ε

+ ε

(3)

(

)

3

4

11 2 1 2

1

1

2

1

1

c

k k k

k c

E

E

a

E

l

h

=

− Ω

⎡

⎤

=

_⎢

+

ω

_⎥

δ +

− ν

− ν

⎣

∑

⎦

(

)

3

2 2

12 2 1 2 2

1

1

2

1

1

c c

k k k k

k c

E

E

a

E

l l

h

=

ν

− Ω

⎡

⎤

ν

=

_⎢

+

ω

_⎥

δ +

− ν

− ν

⎣

∑

⎦

(

)

3

4

22 2 2 2

1

1

2

1

1

c

k k k

k c

E

E

a

E

l

h

=

− Ω

⎡

⎤

=

_⎢

+

ω

_⎥

δ +

− ν

− ν

⎣

∑

⎦

(4)

(

)

3 3 0

1 0 1

1 1 1

1

,

,

k

,

c k k k k k k k

k k k

d

d

r

l

rl

= =

ρ = ρ

− Ω +

∑

ω ρ Ω =

∑

ω ω =

= ω

1k

sin

k

,

2k

cos

k

,

1, 2, 3.

l

=

ψ

l

=

ψ

k

=

(5)

,

,

,

, ,

c c c

E

ν ρ

E

ν ρ

— ,

,

,

,

k k k

E

ω ψ

— ,

(

k

=

1

— ,

k

=

2

— ,

k

=

3

— ),

δ

— ,

h r

( )

—

,

u r

( )

— ,

ω

—

. (5)

.

2

( )

r

ω

— ,

ε

₃

( )

r

—

.

.

) :

( )

2

r

02

const

ε

= ε =

(6)

02

ε

— .

(3)

( )

( )

1

r

3

r

02

ε

= ε

= ε

(7) ,

(

)

2

_{( )}

11 12

11 22

02

0

d a

a

r

a

a

r

dr

+

ω

+

−

+

ϕ

=

ε

(8)

(

)

(

)(

) (

)

2

( )

1

2 2

0 2 1 1 1 0 2 1 2 1 2

r

r

B

= δ ρ

⎢

⎡

r

−

r

+

d

ρ − ρ

r

−

r

+ ρ − ρ

ω

r rdr

+

⎢⎣

∫

(

)

2

_{( )}

2

( )

1 1

3 3 0

13

2

r r

r r

dr

d

h r rdr

l

r

⎤

+

ρ − ρ

⎥

+ ρ

⎥⎦

∫

∫

(9)

(

)

(

)

1 2 1 2

,

2 2 2 1

1

1

c c

c c c c

c c

E

E

σ =

ε + ν ε

σ =

ε + ν ε

− ν

− ν

(

)

(

)

1 2 1 2

,

2 2 2 1

1

1

E

E

σ =

ε + νε

σ =

ε + νε

− ν

− ν

(10)

,

,

(

)

(

)

02 0

1

,

02 0

1

c c c

E

ε ≤ σ

− ν

E

ε ≤ σ

− ν

(11)

1

r

=

r

( )

1 02 1

u r

= ε

r

(12)

, ,

(12). (8) (4), (5)

( )

,

2

( )

,

13

( )

h r

ω

r

l

r

, .

-,

, , (9).

) :

1 01

const

ε = ε =

(13)

2

1 1

2 01

1

,

3 01

1

23

r

r

l

r

r

⎛

⎞

⎛

⎞

ε = ε

_⎜

−

_⎟

ε = ε

_⎜

−

_⎟

⎝

⎠

⎝

⎠

(14)

, ,

1

01 02

2

1

r

r

⎛

⎞

ε

_⎜

−

_⎟

≤ ε

⎝

⎠

(15)

( )

3

r

03

const

ε

= ε =

.

2 01 03

23 1 2

01 1

,

r

l

r

r

r

r

ε − ε

=

≤ ≤

ε

(16)

,

01 2

01 03

1 01 03

,

r

r

ε

ε > ε

≤

ε − ε

(17)

,

N

₁

,

N

₂

( )

1

( )

1

( )

1 1 1

1

c

1

2 2

1

1

r

r

N

f r

a

r

a

h r

r

r

⎡

⎛

⎞

⎤

⎡

⎛

⎞

⎤

=

− δ

_⎢

+ ν

_⎜

−

_⎟

_⎥

ω

+

_⎢

+ ν −

_⎜

_⎟

_⎥

⎝

⎠

⎝

⎠

⎣

⎦

⎣

⎦

( )

1

( )

1

( )

2 2 1

1

c 2 2

1

r

r

N

f

r

a

r

a

h r

r

r

⎛

⎞

⎛

⎞

=

− δ

_⎜

+ ν −

_⎟

ω

+

_⎜

+ ν −

_⎟

⎝

⎠

⎝

⎠

(18)

( )

(

)

1

1 1

1

1

2

3

1

c

1

1 01 1

r

f r

a

E

r

⎡

⎛

⎞

⎤

= δ

− ω − ω

_⎢

+ ν

_⎜

−

_⎟

_⎥

+ δ ε ω +

⎝

⎠

⎣

⎦

(

2

)

2 1

3 01 3

1

23

1

23

r

E

l

l

r

⎛

⎞

+δ ε ω

−

_⎜

−

_⎟

⎝

⎠

( )

(

)

1 2 2 1

2 1

1

1

2

3

1

c 3 01 3 23

1

23

r

r

f

r

a

E

l

l

r

r

⎛

⎞

⎛

⎞

= δ

− ω − ω

_⎜

+ ν −

_⎟

+ δ ε ω

_⎜

−

_⎟

⎝

⎠

⎝

⎠

01 01

1 2 2 2

2

,

1

1

c

c

E

E

a

=

ε

a

=

ε

− ν

− ν

1 1

1c 1

1

c c

,

2c 1

1

c

r

r

a

a

r

r

⎛

⎞

⎛

⎞

σ =

_⎜

+ ν − ν

_⎟

σ =

_⎜

+ ν −

_⎟

⎝

⎠

⎝

⎠

1 1

1 2

1

,

2 2

1

r

r

a

a

r

r

⎛

⎞

⎛

⎞

σ =

_⎜

+ ν − ν

_⎟

σ =

_⎜

+ ν −

_⎟

. (18) (1),

,

ω

₂

( )

r

h r

( )

.

(9).

) .

.

,

2 2 2

1 1 2 2 0

σ − σ σ + σ = σ

1 0 2 0

2

2

cos ,

cos

3

3

3

π

⎛

⎞

σ =

σ

θ σ =

σ

_⎜

θ −

_⎟

⎝

⎠

(19)

1

,

2

ε ε

(

)

0 1

2

1

cos

cos

3

3

E

σ

⎡

⎛

π ⎤

⎞

ε =

+ ν

_⎢

θ − ν

_⎜

θ −

_⎟

_⎥

⎝

⎠

⎣

⎦

(

)

0 2

2

1

cos

cos

3

3

E

σ

⎡

⎛

π

⎞

⎤

ε =

+ ν

_⎢

_⎜

θ −

_⎟

− ν

θ

_⎥

⎝

⎠

⎣

⎦

(20)

(3),

( )

r

θ

:

(

)

1

3

1 2

tg

ln

1

3 tg

d

r

+ν

=

− − ν

θ

d

θ

−

θ

(21)

( )

1

3

tg

1 2

r

θ

= −

− ν

(22)

(20)

ε

₁

( )

r

,

ε

₂

( )

r

( )

( )

2

( )

(

2

)

3

r

1

r l

13 2

r

1

l

13

ε

= ε

+ ε

−

,

( )

( )

( )

03 2

2

13 1 2

1 2

,

r

l

r

r

r

r

r

ε − ε

=

≤ ≤

ε

− ε

(23)

(1) ,

( )

2

r

ω

h r

( )

. ,

(9).

) К . В [4]

(

)

(

2 2

)

1 1 2 2

2

1

2

1

c c c

E

− Ω

ε + ν ε ε + ε +

− ν

(

)

2 3

( )

3

2

2 2 2

1 1 2 2

1

k k k k

k

r

E

l

l

A

r

g

=

ρω ε

+

∑

ω

ε

+ ε

−

= ρ

(24)

(

)

2 3

2 2 2

1 1 2 2

2

2

,

const

1

r

E

A

A

g

ρω ε

ε + νε ε + ε −

= ρ

=

− ν

(25)

(3),

(5) (1)

1

,

2

ε ε

,

13

l

h r

( )

.

В ,

( )

q r

,

( )

1

1 1

r

r

rQ

=

r Q

−

∫

qrdr

=

p r

(26)

( )

1

1 2

dM

r

M

M

p r

dr

+

−

=

(27)

И

M

1

,

M

2

h

δ <

0.1

1 11 1 12 2

,

2 12 1 22 2

M

= ε + ε

b

b

M

=

b

ε +

b

ε

(28)

2

1 1 2 2 1 2 2

1

æ ,

æ , æ

d w

, æ

dw

r dr

dr

ε = −δ

ε = −δ

=

=

(

)

2 3

4

11 2 1 2

1

1

2

3

3

1

1

c

k k k

k c

E

h

E

b

E

l

=

− Ω

⎡

⎤

δ

=

_⎢

+

ω

+

_⎥

δ

− ν

− ν

⎣

∑

⎦

(

)

2 3

2 2

12 2 1 2 2

1

1

2

3

3

1

1

c c

k k k k

k c

E

h E

b

E

l l

=

ν

− Ω

⎡

⎤

δ

ν

=

_⎢

+

ω

+

_⎥

δ

− ν

− ν

⎣

∑

⎦

(

)

2 3

4

22 2 2 2

1

1

2

3

3

1

1

c

k k k

k c

E

h

E

b

E

l

=

− Ω

⎡

⎤

δ

=

_⎢

+

ω

+

_⎥

δ

− ν

− ν

⎣

∑

⎦

(29)

. ,

z

= δ

.

. , , ,

z

= δ

,

1 2 02

ε = ε = ε

.

2 02

1

2

w

= −

ε

r

+

C

(

C

₁ — ).

(

)

(

)

1 11 12 02

,

2 12 22 02

M

=

b

+

b

ε

M

=

b

+

b

ε

(27

(

)

1

11 12 1 1

11 22

02 02

1

r

r

d b

b

r Q

r

b

b

qrdr

dr

+

+

−

=

−

ε

ε

∫

1

Q

—

r

=

r

₁.

( )

,

2

( )

,

13

( )

h r

ω

r

l

r

.

(9).

,

(

01 03

)

2 1

1 01 2 01 13

1

,

C

,

l

r

1

r

C

ε − ε

ε = ε

ε = ε +

=

+

2

01 1

2

2

r

C r

w

= −

ε

−

+

C

δ

δ

1

,

2

C C

— , .

(

)

12

(

)

22

1 11 12 01 1

,

2 12 22 01 1

b

b

M

b

b

C

M

b

b

C

r

r

=

+

ε +

=

+

ε +

(26)

( )

1 12 11 22 1 12 22

11 12 2

01 01 01

p r

C b

b

b

C b

b

d

b

b

dr

r

r

r

r

⎛

⎞

−

−

+

+

+

+

=

⎜

_ε

⎟

_ε

_ε

⎝

⎠

( )

,

2

( )

h r

ω

r

.

.

- ,

.

[5].

1. . . . / . XXI

. . . : - , 2005.

. 255–264.

2. . . .

. : - -

, 2002. . 241–249.

. // . 2002. . 8. № 3.

4. Nemirovsky Yu. V. Optimum design of three-layered inhomogeneous shell and plates. Proc. 6th Russian-Korean Intern. Sympos. on Science and Technology “KORUS– 2002”. Novosibirsk, 2002. Vol. 2. P. 15–19.

5. . ., . .

. . // .

8- . . . : - , 2006. . 23–29.