Volume 6, 2002
1
THE BOUNDARY VALUE PROBLEM IN DISPLACEMENTS FOR
CUSPED PRISMATIC SHELLS ON THE CASE OF THE FIRST
APPROXIMATION OF I. VEKUA’S MODEL
G. Devdariani
I. Vekua Institute of Applied Mathematics
I. Javakhishvili Tbilisi State University
In the case of the first approximation of I. Vekua’s version of the elastic shell
theory the general system consists of six second order differential equations.
This system is split into two autonomous systems. One of them has the form
!
!
! 























∂
∂
∂
∂v1
∂v1
∂v2
h3
+µ
h3
+λ
h3
(λ + 2µ)
∂x
∂x
∂y
∂y
∂x
∂y
!
∂u3
∂
∂v2
h3
− µh
− 3µhv1 + Y1 = 0,
+µ
∂y
∂x
∂x !
!
!
∂
∂
∂
3 ∂v2
3 ∂v2
3 ∂v1
h
+ (λ + 2µ)
h
+µ
h
µ
∂x
∂x
∂y
∂y
∂x
∂y
!
∂u
∂
∂v
3
1
+λ
h3
− µh
− 3µhv2 + Y2 = 0,
∂y
∂x
∂y
!
!
∂u3
∂u3
∂
∂
∂
∂
h
+µ
h
+ 3µh (hv1 ) + 3µh (hv2 )
µ
∂x
∂x
∂y
∂y
∂x
∂y
+Y3 = 0,























(1)
(The second one was investigated ealier and this article is a continuation of
the paper [2]), where v1 , v2 and u3 are unknown functions and, namely, they
are so called first (v1 , v2 ) and zero (u3 ) moments of the displacement vector; λ
and µ > 0 are Lame’s constants; Y1 , Y2 and Y3 are the first (Y1 , Y2 ) and zero
(Y3 ) Fourier-Legendre moments of the given volume force, 2h is the thickness
of the shell. The system is considered on the so called middle surface ω of
the shell, which is actually the orthogonal projection of the shell on the plane
Oxy.
In this paper we study the case when the middle surface ω is a plane
bounded domain with a smooth boundary ∂ω ∈ C 1 , where
∂ω = Γ0 ∪ Γ1 ,
Γ0 = {(x, 0) : a ≤ x ≤ b, a, b ∈ R, a < b} ,
Γ1 = {(x, y) : (x, y) ∈ ∂ω, y ≥ 0} ,
Γ0 ∩ Γ1 = {(a, 0); (b, 0)} .
Let the thickness 2h be given by the function
h = h(y) = y m , m > 0.
2
Bulletin of TICMI
Let D(ω) be a set of infinitely differentiable compactly supported functions
on ω. We introduce the following bilinear form on [D(ω)]3 :
(u, v)3,m ≡
+
Z
y
ω
m
3
Z
y
m
ω
" "
∂u1 ∂u2
∂u1 ∂v1 ∂u2 ∂v2
+
+
+
∂x ∂x
∂y ∂y
∂y
∂x
∂u3
+ 3u1
∂x
! !
! ∂v1 ∂v2
+
∂y
∂x
! ∂v3
+ 3v2
∂y
∂u3
∂v3
+ 3v1 +
+ 3u2
∂x
∂y
!#
!#
dτ
dτ, (2)
where u = (u1 , u2 , u3 ), v = (v1 , v2 , v3 ) ∈ [D(ω)]3 .
According to the Korn’ weighted inequality [see [2], p.35] it is easy to show
that (2) is a scalar product on [D(ω)]3 . If we complete [D(ω)]3 by the scalar
0
product (2) we obtain the Hilbert space M
kuk23,m
+
Z
ω
=
Z
y
m
ω
 !2
∂u
1

∂x
∂u3
y 
+ 3u1
∂x
m
3
!2
∂u2
+
∂y
m (ω)
!2
with the norm
∂u1 ∂u2
+
+
∂y
∂x
∂u3
+ 3u2
+
∂y
0
!2 
 dτ,
!2 
 dτ
(3)
where u = (u1 , u2 , u3 ) ∈ M m (ω).
Let
1 ) (ω) := L2,σ (ω) × L2,σ (ω) × L2,σ 1 (ω),
L2,(σm ,σm ,σm
m
m
m
where L2,σm (ω) is a Hilbert space of measurable functions ψ such, that the
norm
1

kψkL2,σm (ω) =
is finite,
σm (y) =




Z
2
σm (y)ψ
ω
y 1−ε , ε > 0, m = 1,
1
σm
= y 4−m .
0
B(v, w) :=
y
ω
3m
"
(λ + 2µ)
#
(x, y)dτ 
y 2−m , m =
6 1,
We say that v = (v1 , v2 , u3 ) ∈ M
system (1) if
Z (
2
m (ω)
is a generalized solution of the
∂v1 ∂w1
∂v2 ∂w1
∂v1 ∂w1
+µ
+λ
∂x ∂x
∂y ∂y
∂y ∂y
"
∂u3
∂v2 ∂w1
+µ
+ ym µ
w1 + 3µv1 w1
∂x ∂y
∂x
+y
3m
"
#
∂v2 ∂w2
∂v2 ∂w2
∂v1 ∂w2
∂v1 ∂w2
µ
+ (λ + 2µ)
+µ
+λ
∂x ∂x
∂y ∂y
∂y ∂x
∂x ∂y
#
Volume 6, 2002
+y
m
3
"
∂u3
1 ∂u3 ∂w3 1 ∂u3 ∂w3
∂w3
∂w3
µ
w2 + 3µv2 w2 + µ
+ µ
+ µv2
+ µv1
∂y
3 ∂x ∂x
3 ∂y ∂y
∂y
∂x
=
Z ’
#)
dτ
“
1
Y1 w1 + Y2 w2 + Y3 w3 dτ
3
ω
for all w = (w1 , w2 , w3 ) ∈ [D(ω)]3 .
Theorem. If Y = (Y1 , Y2 , Y3 ) ∈ L2,(σ3m ,σ3m ,σ1 ) (ω) then the system (1)
3m
0
has a unique generalized solution v = (v1 , v2 , u3 ) ∈ M
the estimate
,
kvk3,3m ≤ ckY kL
2,(σ3m ,σ3m ,σ 1 )
3m
3m (ω)
and there holds
where c is a positive constant independent of Y and v.
0
Proof. Let us show that the form B(v, w) is coercive on M
0
B(v, v) :=
ω



∂v1
y m y 2m (λ + 2µ)

∂x
 ∂u3
∂v2
+µ
v1 +3µv12 +y 2m µ
∂x
∂x
!2
1
∂u3
∂u3
v2 + 3µv22 + µ
+µ
∂y
3
∂x
=
Z
ω
 ∂v1 ∂v2
y 3m λ
+
∂x
∂y
!2
!2
 ∂v1
+ 2µ 
∂x
≥µ
ω
+δ
y 3m 
Z
ω
where 0 < δ ≤
Hence,
1
µ.
3
∂v1
∂x
 !2
!2
!2
  ∂v1
+µ
∂y
1
∂u3
+ µ
3
∂y
1 Z m  ∂u3
+ µ y
+ 3v1
3 ω
∂x
Z
∂v2
+ (λ + 2µ)
∂y
!2
+
!2
!2
∂v2
∂y
∂u3
+ 3v1
ym 
∂x
!2
!2

∂v2 ∂v1
∂v2 ∂v1 
+λ
+µ
∂y ∂x
∂x ∂y

∂v1 ∂v2
∂v1 ∂v2 
+µ
+λ
∂y ∂x
∂x ∂y

∂u3
∂u3 
+ µv2
+ µv1
dτ
∂x
∂y 
∂v2
+
∂y
!2 
∂v1
+µ
∂v2
+
∂y
∂x
∂u3
+
+ 3v2
∂y
!2
First
3m (ω).
we estimate B(v, v) for v = (v1 , v2 , u3 ) ∈ M
Z
3m (ω).
+
!2 
!2 
∂v2 
dτ
∂v1
+
∂y
∂x
∂u3
+
+ 3v2
∂y
B(v, v) ≥ c1 kvk23,3m .
 dτ
!2 
 dτ,
!2 
 dτ
4
0
M
Bulletin of TICMI
Let us estimate |B(v, w)|, where v = (v1 , v2 , u3 ), w = (w1 , w2 , w3 ) ∈
3m (ω).
| B(v, w) |:=
Œ
ŒZ
Œ
Œ
Œ
Œ
y
3m
ω
"
∂v1 ∂v2
∂v1 ∂w1
(λ + 2µ)
+µ
+
∂x ∂x
∂y
∂x
! ∂w1 ∂w2
+
∂y
∂x
!
#
∂v1 ∂w2
∂v2 ∂w1
∂v2 ∂w2
+λ
+ (λ + 2µ)
+λ
dτ
∂y ∂x
∂y ∂y
∂x ∂y
"
Z
∂u3
∂w3
∂u3
µ
∂u3 ∂w3 ∂u3 ∂w3
+
+3
w1 + 3
w2 + 3v1
ym
+
3ω
∂x ∂x
∂y ∂y
∂x
∂y
∂x
#
Œ
Œ
Œ
+λ
∂v2 ∂w1
∂v2 ∂w2
+ (λ + 2µ)
∂y ∂x
∂y ∂y
"
Z
Œ
Œ
∂w3
∂v1 ∂w1
+3v2
+ 9v1 w1 + 9v2 w2 dτ ŒŒŒ = ŒŒ y 3m (λ + 2µ)
∂y
∂x ∂x
Œ
ω
∂v1 ∂v2
+µ
+
∂y
∂x
! !
∂w1 ∂w2
+
∂y
∂x
#
µZ m
∂v1 ∂v2
dτ +
y
+λ
∂x ∂y
3ω
∂u3
+
+ 3v2
∂y
" ! ∂u3
+ 3v1
∂x
∂w3
+ 3w2
∂y
! !#
Hence, if we use the Hölder inequality, we obtain
∂w3
+ 3w1
∂x
!
Œ
Œ
dτ ŒŒ .
Œ
|B(v, w)| ≤ c2 kvk3,3m kwk3,3m .
It is evident that (see [2], Lemma1, p.28)
Œ
ŒZ
Œ
Œ
Œ
Œ
ω
Y
Œ
Œ
Œ
· wdτ ŒŒ
Œ
≤ c3 kwk3,3m kY kL
(
2, σ3m ,σ3m ,σ 1
3m
)
.
Thus, we have shown, that B(v, w) is coercive and bounded, and the funcR
0
tional Y · wdτ is bounded on M
ω
3m (ω).
Now, the Lax-Milgram theorem (see,
e.g. [3]) completes the proof.ƒ
In what follows we make some ramarks concerning the theorem.
0
At first we remark, that ∀u ∈ M
3m (ω)
u|Γ1 = 0,
and
0
M
3m (ω)
0
⊃H
2,3m (ω)
0
(Definition and properties of space H
0
×H
2,3m (ω)
2,m (ω)
0
×H
1,3 (ω)
one can see in the paper [2]).
Volume 6, 2002
0
Let v = (v1 , v2 , u3 ) ∈ M
If 0 < m < 31 , then
3m (ω)
5
be a generalized solution of system (1).
v1 |∂ω = v2 |∂ω = u3 |∂ω = 0
in the sense of the trace.
Consequently in this case v must be given on the whole boundary if we
consider boundary conditions in displacements.
If 13 < m < 1, then
v1 |Γ1 = v2 |Γ1 = 0, u3 |∂ω = 0.
v1 and v2 do not have traces on Γ0 , in general. In this case v1 and v2 must be
given only on Γ1 , and u3 must be given on the whole boundary.
If m ≥ 1, then
v1 |Γ1 = v2 |Γ1 = u3 |Γ1 = 0
and v1 , v2 and u3 do not have traces on Γ0 , in general. Therefore, we must
give v1 , v2 and u3 only on Γ0 , and have them free from boundary conditions
on Γ1 .
REFERENCES
[1] Vekua I.N., Shell Theory: General Methods of Consruction. Pitman advanced publishing program. Boston-London-Melburne, 1985
[2] Devdariani G., Jaiani G., Natroshvili D., Kharibegashvili S., Some Mathematical
Problems Related to the First Approximation of I. Vekua’s Theory for Cusped Prismatic
Shells. Appl.Math.Inf., 5, 2, 2000, 26-46
[3] Fichera G., Existence Theorems in Elasticity. Handbuch der Physik. Band 6a/2,
Springer-Verlag, Berlin, Heidelberg, New-York, 1972.
Received September 3, 2002; revised November 15, 2002; accepted December 3, 2002.