20
Ի Ի ՈՒԹ ՈՒ Ի Ի Ի Ի Ղ Ի
В АЦ А Ь А А А А
60, №4, 2007
539.3, 624.04
Ч Ч ,
Ы Щ
. ., . .
лючевыеслов : , , , , .
Key words: half-plane, opening, crack, tension, displacement.
. . , .
և
և
: և
, և
: p
( -1):
A.H. Babloyan, A.V. Baghdasaryan
Symmetric Problem of Elasticity Theory for a Half-Plane Weakened with a Round Opening and an Internal Crack
The article presents the solution of a symmetric problem of elasticity theory for an elastic half-plane weakened by a round opening and a rectilinear internal crack, the latter being perpendicular to the edge of the half-plane. Symmetrically distributed normal loadings are given at the edges of the opening, the half-plane and banks of the split. On the infinity the half-plane spreads by equally distributed loadings with p intensity (fig.1).
.1
,
,
.
с0.
,
.
p ( .1).
[1].
sin
sh
,
,
cth
ch
cos
ch
cos
2
a
a
i
x
=
α
y
=
α
x
+ =
iy
ai
α + β
α −
β
α −
β
(1):
1 2
(0, )
( ),
( , )
( ), (0
)
(0, )
0,
( , )
0, (0
),
( , )
0, (0
)
a
αf
a
αf
αβ αβ αβ
σ
β =
β
σ γ β =
β
≤ β ≤ π
τ
β =
τ γ β =
≤ β ≤ π
τ α π =
≤ α ≤ γ
(2)0 0 0 0
( , )
( ), (
),
v( , )
v , (0
)
a
σ α π =
βf
α
c
≤ α ≤ γ
α π =
≤ α ≤
c
(3)Э
0
1 0
( , )
( , )
m( ) cos
m n( ) cos
m n
g
a
b
n
∞ ∞
= =
21 0
Φ
,a
m( )
β
b
n( )
α
,[–3].
, ,
(4) :
1 2
2,4,6 0
1 2
1,3,5 0
8
( 1)
[
( )
( )]cos
d
( , )
(0)
( )
( 1)
(
1)
cth
(0)
( )
(
1, 2,...)
2
1
2
8
( 1)
[
( )
( )]cos
d
( , )
(0)
( )
( 1)
(
th
2
1
n
m m
n n
m
n
n n
n n
n
m m
n n m
n
n n
Z
X
w
w
n
m n
d
d
n
d
d
n
n
Z
Y
w
w
n
m n
d
d
n
n
π ∞
=
π ∞
=
δ
χ
+
= −
β +
β
β β −
γ∆
−
γ
π −
⎡
+ γ
⎤
−
⎢
+
+
γ
⎥
=
+
⎣
⎦
δ
η
+
= −
β −
β
β β −
γ∆
+
γ
π −
+
−
+
∑
∫
∑
∫
1)
(0)
( )
(
1, 2,...)
2
d
nd
nn
γ
⎡
+
−
γ
⎤
=
⎢
⎥
⎣
⎦
(5)
1 2
1 2
4 2 2 2 2
sh(
)
sh
1,
,
(
2, 3,...)
(
1)[ch(
)
ch ]
th
sh(
)
sh
,
,
(
2, 3,...)
4
(
1)[ch(
)
ch ]
( , )
2(
1)
(
1)
n
n
m m
n
n
n
n
n
n
n
n
n
n
m n
n
n
γ +
γ
χ =
χ =
=
−
γ −
γ
γ − γ
γ −
γ
η =
η =
=
−
γ +
γ
∆
= δ +
+ δ +
−
(6)
0
X
,Y
0[
]
(
0 0 0)
1 2
2,4,6 0
(0)
( )
ch
1
cth( / 2)
8
[
( )
( )]d
( , )
m m
m
d
d
X
Z
w
w
m n
π ∞
=
π
+
γ +
γ + + γ
γ
+
δ
+
=
β +
β
β
γ∆
∑
∫
[
]
(
0 0 0)
1 2
1,3,5 0
(0)
( )
ch
1
th( / 2)
8
[
( )
( )]d
( , )
m m
m
d
d
Y
Z
w
w
m n
π ∞
=
π
−
γ +
γ − − γ
γ
+
δ
+
=
β −
β
β
γ∆
∑
∫
(7)(5)–(7) [3]. (5)
2 /
π
. (3). [4,5]
.
,...)
2
,
1
(
,
2
,1
=
=
+
∑
∞=
m
b
A
a
A
mk k mk
m (8)
0.5
, ,
1
(cos ) (cos ) ctg
d ,
0(
)
2
m k k m k m k
k c
a
kN
z
z
a
m
π ∞
− =
θ
22
1 2
0
1 1
1 2
1 2
0 0
( )
(cos ) ctg
d
( )
(cos ) ctg
d
2
2
( )
( )
( ),
( )
( )
( )
( ) sin( / 2) d
( ) sin( / 2) d
2 2
2 2
( )
,
( )
cos
cos
cos
cos
c
m m m
c
m m m m m m
b
m F
z
m F
z
z
x
P x
P
x
y
x
P x
P
x
g x
x
x
g z
x
x
F
F
x
x
π
+ +
θ π
θ
θ
=
θ
θ
θ −
θ
θ
θ
=
−
=
+
′
θ =
θ
π
−
θ
π
θ −
∫
∫
∫
∫
(10)
2
0
1 0 0
2
2 0 0
1
2
2 v
,
( )
sh
[ ( )
( )]
1
2
(1
)
( )
( , )
( )
( 1)
( )
m m
m
m
n n n
Z
A
g x
b
b
g x
w
b
∞
=
δ
′
µ
π
′
′′′
=
= −
α +
α +
α
− ν
γ + δ
′′
′′
′′
= Φ α π −
α +
∑
−
α
(8) :
1 2
1 1
1
1 0
( ) ctg( / 2) d
cos(
)
2
(0)
2 sin
2
cos
cos
(cos ) ctg( / 2) d
( ) ctg( / 2) d
(
)
cos
cos
cos
cos
x
m
m c
c k
k k
k c
F
x
m A
mx
g
x
z
F
A N
c
x
x
x
∞ − =
π ∞
=
⎧
θ
θ
θ
=
−
⎨
+
θ −
⎩
⎫
θ
θ
θ
θ
θ
θ
+
−
⎬
≤ ≤ π
θ −
θ −
⎭
∑
∫
∑
∫
∫
(11)
1/ 2 1
1
( ) d
(1
) cos(
)
2
cos
2
cos
cos
cos
cos
c I
m m
m x
K
F
x
A
N
mx
x
x
∞
− =
⎧
′ θ θ
⎪
+
=
⎨
−
−
+
−
θ
−
θ
⎪⎩
∑
∫
2 2
2 1
(cos ) tg( / 2) d
( ) ctg( / 2) d
( )
cos
cos
cos
cos
c k
k k
k x c
y
F
k A N
g
x
x
π ∞
=
⎫
′
θ
θ
θ
+
θ
θ
θ⎪
+
π
⎬
−
θ
−
θ ⎪⎭
∑
∫
∫
(12)–
)
(
)
(
cos
1 21
c
F
c
F
c
N
A
k
K
k kk
I
=
∑
−
+
∞
=
(13)
(8) [4,5]. (11) (12)
(
0
≤
α
<
c
)
v(
α
,
0
)
0(
c
≤ α ≤ γ β =
,
0)
.1. . . .
: , 1967. 402 .
2. . . . .: , 1950.
3. . ., . . ,
. // . . .
2007. .60. №3.C.15–22.
4. . ., . .
. .: , 1983.488 .
5. . . .
/ . – : 1979, 346 .
