Ա ԱՍ Ա ՈՒԹ ՈՒ Ա Ա Ա Ա Ա Ղ Ա
61, №1, 2008
539.3
. .
Ключевыеслова: ,
r
−1,, .
Keywords: elastic wedge, stress singularities in form
r
−1, neighborhood of angular point, transcendental equation.Ա. .Ս Ա
ք
α
ք(
0
<
α
≤
2
π
)
ք ,
:
, և 1969 ., Ա. .
α
<
π
ք ,π
≤
α
≤
2
π
ք :π
=
α
ևα
=
2
π
ք:
A.M.Sargsyan
On Stresses Singularity in one Problem of Elasticity Theory for the Wedge
The behavior of the stresses in the angular point neighborhood of the elastic wedge with arbitrary opening
angle
α
(
0
<
α
≤
2
π
)
is considered. On the radial bounds of the wedge the boundary conditions of thesmooth contact are given.
The investigation of the transcendental equation roots at
α
<
π
, determining the order of the stressessingularity , which was carried out by A.I.Kalandia in 1969, is completed for the interval
π
≤
α
≤
2
π
.The mechanical sense of the stresses singularity of
r
−1 type in case ofα
=
π
andα
=
2
π
is interpreted.
(
<
α
≤
π
)
α
0
2
, o.
,
, 1969 . . .
α
<
π
,π
≤
α
≤
π
2
.1 −
r
α
=
π
,π
=
α
2
.,
α
, ( I-I)
( II-II), , –
( I-II), , - , . . [1].
∑
= +
λ
λ
−
−
ϕ
+
λ
−
−
ϕ
=
ϕ
Φ
21
1
[
cos(
(
1
)
)
sin(
(
1
)
)
]
)
,
(
j
j j
j
j
B
A
r
r
λ
,
,
jj
B
A
– ,,
λ
.α
,ν
,.
[2] . . , –
, ( I, III
I-II) III-III,
0
),
(
τ
=
=
ϕϕ
f
r
ru
.0
sin
sin
λα
±
α
=
. , (1)
0
),
(
σ
=
=
f
r
ϕu
r.
[3].
– I-III
II-III ( ).
3
,
0
=
ν
α
0
<
α
≤
2
π
.λ
Re
min
.Re
λ
<
1
λ
−
min
Re
1
., III-III, . . : « III-III
)
,
0
(
π
.λ
=
0
,α
=
π
,».
. . :
1. К я ы
α = π
?
α
=
π
(1)λ
=
k
(
k
=
0
,
1
,
2
,
…
)
,0
=
λ
,. :
π
≤
α
<
0
(1)λ
min=
π
α
−
1
α
=
π
−
ε
(ε
–)
)
(
1
−
ε
π
−
ε
.ε
→
0
r
1
.. .- . . . . ,
[4, 5].
. 1 . 2
1 2 0
2 2 2 2
[(
)
]sgn
( )
(
)(
)
P
P x
c
x
p x
a
x
x
b
+
−
=
π
−
−
(2))
(
x
p
–
,)
(
)
(
2
1 20
P
P
k
K
a
c
−
′
π
=
(3)0
c
–,
)
(
k
K
′
–,
a
x
b
a
b
k
′
=
1
−
2 2,
<
<
)
)(
(
)
(
sgn
)
(
2 2 2 2 3b
x
x
a
k
K
x
aP
x
−
−
′
=
τ
(2'))
(
x
τ
–
)
(
)
(
2
3k
K
k
K
E
P
′
=
δ
(3'
)δ
– ,a
x
b
a
b
k
=
,
<
<
∫
−
−
′
=
′
10
(
1
2)(
1
2 2)
)
(
t
k
t
dt
k
K
( )
(
b
→
0
)
,)
(
x
p
(τ
(
x
)
)
1
r
( . 1 2).,
P
j(
j
=
1
,
2
,
3
)
,,
δ
c
0 (K
(
k
′
)
0
→
b
).0
c
δ
, , (3) (3'
),P
j.
r
−1.
(2
'
) (3'
) . ..
2. П х ы III–III
π
≤
α
≤
π
2
, я я яπ
>
min
1,
0
1
,
3
2
2
1, 3
2
2
π α −
< α ≤ π
⎧
⎪
λ
=
⎨
− π α
π ≤ α ≤ π
⎪ π α −
π ≤ α ≤ π
⎩
min
λ
α
.3.π
=
α
2
λ
min=
0
, . . (r
→
0
)1 −
r
., [6, 7]
,
,
min
Re
λ
[2] - .
, III-III
π
≤
α
≤
π
2
,
,
π
=
α
2
., [8]
}
2
,
,
0
{
0 01
≤
≤
−
α
≤
ϕ
≤
α
α
>
π
Ω
r
a
Ω
2{
a
≤
r
<
∞
,
−
α
0≤
ϕ
≤
α
0}
( . 4),
,
0
=
τ
rϕϕ
=
±
α
0(
α
=
2
α
0)
:
)
u
ϕ(
r
,
±
α
0)
=
±
g
1(
r
)
(
1
≤
r
<
∞
)
,)
σ
ϕ(
r
,
±
α
0)
=
f
2(
r
)
(
1
≤
r
<
∞
)
,u
ϕ(
r
,
±
α
0)
=
±
g
2(
r
)
(
0
≤
r
≤
1
)
) ),
I-I III-III, .
,
,
– e ,
[9]. ,
a
2
Ω
Ω
10
α
=
ϕ
0 = ϕ
P
. 4
π
α
2
π
0 1 2
min
λ
r
a
r
u
ϕ(
,
±
α
0)
=
0 .. ,
, , (1).
α
0=
π
,. .
r
−1,,
, « « »
». [10]
,
(
x
<
b
),
( . 5).
.
. 5
:
)
(
0
)
0
,
(
),
(
)
(
)
0
,
(
x
f
x
x
b
x
b
x
a
u
z=
±
<
τ
yz=
<
<
± ±
,
–
a
x
x
b
x
x
f
x
u
±y(
,
0
)
=
±
(
)
(
<
),
τ
±yz(
,
0
)
=
0
<
,)
(
0
)
0
,
(
x
x
b
y
=
<
σ
±.
.
)
0
,
(
x
yzτ
x
<
b
,x
>
a
,–
σ
y(
x
,
0
)
x
<
b
x
>
a
. ,(3) (3
'
). ,b
→
a
,
r
−1.,
r
−1, , ,
.
. .
1 −
r
,c
0δ
, : ,( ) ,
0
=
x
σ
x,j
P
,P
j, ,, . ,
[10], ,
y
x
a
b
−
a
b
2
(b
=
a
),.
0
→
b
( . 1, 2)b
→
a
( . 5),
, , – ,
.
1. Williams M. L. Stress singularities resulting from various boundary conditions in angular corners of plates in extensions. // J. Appl. Mech. 1952. Vol. 19. № 4. P. 526-528.
2. . . .
// . 1969. .33. № 1. . 132-135.
3. Melan E. Ein Beitrag zur Theoric geschweisster Verbindungen. // Ing.- Archiv. 1932. Bd. 3. Heft 2. p.123.
4. . . . .– .: ,
1949. 270 .
5. . ., . .
. .: , 1983. 487 .
6. . ., . . . .: ,
1977. 312 .
7. . ., . . . .: ,
1981. 688 .
8. . .
. // . . . 1990. .43. № 2. . 12-21.
9. . ., . .
. // . . . 1990. .43. № 2. . 3-11.
10.Gevorgyan S.Kh., Manukyan E.H., Mkhitaryan S.M., Mkrtchyan M.S. On A Mixed Problem For An Elastic Space With A Crack Under Antiplane And Plane Deformations. Collection of Papers. Yerevan-2005. 282p.
