Volume 8, 2004
1
THE METHOD OF A SMALL PARAMETER FOR THE
SHALLOW SHELLS
T. Meunargia
I. Vekua Institute of Applied Mathematics
of I. Javakhishvili Tbilisi State University
Abstract. In the present paper the method of a small parameter is used
for the shallow shells of I.Vekua. The small parameter has the form ε = Rh ,
where h is a semithickness of the shell and R is the characterictic radius of
curvature of the midsurface of the shell.
Key words: Shallow shells, small parameter.
MSC 2000: 74K25
1. Introduction
A complete system of three-dimentional equations of elastic bodies of shallow shell type can be written as [1]:
a) equilibrium equations

3β

 ∇ασ αβ − bβα σ α3 + ∂σ + Φβ = 0,
∂x333
(1)

 ∇ασ α3 + bαβ σ αβ + ∂σ + Φ3 = 0, (α, β = 1, 2)
∂x3
where σ ij (i, j = 1, 2, 3) are contravariant components of the stress tensor,
bαβ (bαβ , bβα) are covariant (contravariant, mixed) components of the curvature
tensor of the midsurface S of the shell Ω, Φi (i = 1, 2, 3) are contravariant components of the external force, ∇α (α = 1, 2) are symbols of covariant derivatives, x1 and x2 are curvilinear coordinates of the midsurface, x3 is thickness
coordinate, −h ≤ x3 ≤ h, and h is a semi-thickness of the shell;
b) Hooke’s law


σ αβ = λ(θ − 2HU3 + ∂3 U3 )aαβ + µ(∇αU β + ∇β U α − 2bαβ U3) = σ βα,



 σ α3 = µ(∂ 3U α + ∇α U 3 + bαγU γ ) = σ 3α,
(2)
σ 33 = λ(θ − 2HU3 ) + (λ + 2µ)∂3 U3 ,


∂


 (θ = ∇γ U γ , ∇α = aαγ ∇γ , ∂ 3 = ∂3 =
, aαβ = rαrβ ),
∂x3
where λ and µ are Lamé constants, H is the mean curvature of the surfaces
2H = b11 + b22 = bαα,
2
Bulletin of TICMI
U = uαrα + U 3 n is the displacement vector, rα(rα )(α = 1, 2) are covariant
(contravariant) base vectors and n is a normal of the midsurface S;
c) the stress vector σ (l) acting on the lateral surface with normal l has the
form [1]
σ (l) = σ α lα == (σ αβ rβ + σ α3n)lα = σ(ll)l + σ(ls)s + σ(ln)n ⇒
σ(ll) = σ αβ lα lβ , σ(ls) = σ αβ lαsβ , σ(ln) = σ α3lα ,
(lα = lrα, sα = srα , l × s = n).
(3)
There exist several methods of reduction of the three-dimensional problems to
the two-dimensional problems (Kirchhoff-Love, E. Reissner, K. Friedrichs, A.
Green, A. Goldenveizer, I. Vorovich, I. Vekua, etc.)
2. I. Vekua’s Demension Reduction Method
Following I. Vekua we assume the validity of the expansions
∞ x (m) ij (m) i (m) i
3
ij
i
i
σ ,U , Φ
⇒
Pm
(σ , U , Φ ) =
h
m=0
h
x (m)
(m)
(m)
2m + 1
3
ij
i
i
σ ,U ,Φ
dx3 ,
(σ ij , U i , Φi )Pm
=
2h
h
(4)
−h
where Pm are Legendre polynomials of order m.
Substituting the above expansions in relations (1), (2) and (3) having satisfied beforehand the conditions on the face surface x3 = ±h,
(±) 3
σ 3 (x1, x2 , ±h) = σ
,
we obtain the following infinite complete system of two-dimensional equations,
which in the izometric coordinates
ds2 = Λdzdz̄ (z = x1 + ix2, Λ(x1, x2 ) > 0)
written in a complex form looks as follows [2]:
a) equilibrium equations:

(m)


(m)
(m)
(m)
h ∂ (m)
∂ σ αα


σ 11 − σ 22 + 2i σ 12 + h
− εHR σ +


Λ ∂z
∂ z̄



(m)

(m)
(m−1)
(m−3)


σ
σ
σ
−εQR
−
(2m
+
1)(
+
+
...)
+
h
F+=0

+
+
+



 

(m)
(m)
(m)
h ∂ σ + ∂ σ + 



+
+ εHR σ αα


Λ
∂z
∂ z̄





(m)
(m)
(m)


+εRe[QR( σ 11 − σ 22 + 2i σ 12 )]




(m)
(m−1)
(m−3)

−(2m + 1)( σ 33 + σ 33 + ...) + h F + = 0 (m = 0, 1, ...),
(5)
Volume 8, 2004
where ε =
of S.
Then
h
R
3
is a small parameter and R is the characteristic radius of curvature
∂
1
=
∂z
2
(m)
σ+
∂
∂
−i 2
1
∂x
∂x
(m)
1
∂
=
∂ z̄
2
,
(m)
(m)
= σ 1 + i σ 2, F
∂
∂
+i 2
1
∂x
∂x
(m)
,
(m)
= F 1 + i F 2,
(m)
(m)
(−)
2m + 1 (+)
2Q = b11 − b22 + 2ib21, F i = Φ i +
[ σ 3i − (−1)m σ 3i].
2h
b) Hooke’s law

(m)
(m)
(m)
(m)
(m)
∂
1


h( σ 11 − σ 22 + 2i σ 12) = 4µΛ h
U + − εQR U 3 ,


∂z̄ Λ



(m)
(m)
(m)  (m)α


σ
−
2HεR
=
2(λ
+
µ)
h
+
2λ
h

θ
U
U 3,
3
α



(m)

(m)
(m)
(m) (m)
∂U3



h σ + = µ 2h
+ εR(H U + + Q U + ) + U +  ,


∂ z̄





(m)
(m)
(m)  (m)
h σ 33 = λ(h θ − 2HεR U 3) + (λ + 2µ) U 3 ,
where
(m)
(m)
(m)
(m)
U + = U 1 + i U 2,


(m)
(m)
(m)
1 ∂ U + ∂ U +
+
,
θ = 
Λ
∂z
∂ z̄
U
(m)
U
α
+
(6)
(m)
= U rα, (α = 1, 2),
= (2m + 1)
(m+1)
(m+3)
U + U + ... ;
c) the boundary conditions for stress tensor on the lateral contour Γ:
(m)
σ (ll)
(m)
= f 1,
(m)
σ (ls)
(m)
= f 2,
(m)
σ (ln)
(m)
= f 3, ⇒

2 (m)
(m)
(m)
(m)
(m)
(m)

1
dz̄


σ (ll) + i σ (ls) =
σ αα − ( σ 11 − σ 22 + 2i σ 12)


2
ds




(m)
(m)
= f 1 + i f 2,




(m)


(m) dz̄
 (m)

 σ (ln) = −Im σ +
= f 3,
ds
(m = 0, 1, ...).
Substituting (6) in (5) and using the formula


(m)
(m)
1 ∂ ∂ 1 (m)
2 ∂  1 ∂ U +
Λ
+ 2ε2 KR2 U + ,
4h2
U + = 4h
Λ ∂z ∂ z̄ Λ
∂ z̄ Λ ∂z
(7)
4
Bulletin of TICMI
where K is the Gaussian curvature of the midsurface S, we obtain a system
of differential equations in terms of the displacement vector components:



(m)
(m)
(m) 1
∂
∂
∂
∂
U
θ
U 3
+
 4µh2 
+ 2(λ + µ)h2
+ 2λh

∂ z̄ Λ ∂z
∂ z̄
∂ z̄







(m−1)
(m−3)

(m−1)
(m−3)

 −(2m + 1)µ2h ∂ U 3 + U +2h ∂ U + + U + ...
+
+

∂ z̄
∂ z̄



(m)
(m)



 −εR 4µh ∂ΛQ U 3 + 4(λ + µ)h ∂H U 3

 Λ
∂z
∂ z̄








(m)
(m)
(81 )

 +µ H 2h ∂ U 3 + (m)  + Q 2h ∂ U 3 + (m) 
U +
U +

∂ z̄
∂ z̄





(m−1)
(m−1)

 −(2m + 1)µ
H U + + Q U + + ...





(m)
(m)
(m)

+ε2 R2 µ (2K − H 2 − QQ̄) U + − 2HQ U + + h2 F + = 0,



























2
2
(m)
(m)
µ h ∇ U 3+h θ
(m−3)
+ λh θ
− (2m + 1)
(m−3)
+ (λ + 2µ) U


 2µh
(m)
3
λh θ
(m−1)
+ (λ + 2µ) U
3
+ ...
(m)
∂(H U + + Q U + )
Re
+εR

∂z
 Λ
(m)
(m−1)
(m)
(82 )
∂
+H 2(λ + µ)h θ + 2λ U 3 + 4µhRe Q
∂ z̄
(m−1)
(m−3)
+ 2(2m + 1)λH
U 3 + U 3 + ...
1 (m)
U+
Λ
(m) (m)
−4ε2 R2 (λ + µ)H 2 + µQQ U 3 h2 F 3 = 0,
4 ∂2
2
.
m = 0, 1, ...; ∇ =
Λ ∂z∂ z̄
The passage to finite systems is performed by considering a finite expansion
(4), where m = 0, 1, ..., N.
Volume 8, 2004
5
3. Approximation of order N = 0
By introducing the following notation
(0)
(0)
U i = Ui ,
(0)
σ ij
(U
i
= 0)
(0)
= Tij ,
F i = Xi ,
we obtain:
a) equilibrium equations

∂
h ∂

(T11 − T22 + 2iT12) + h Tαα − εR(HT+ + QT + ) + hX+ = 0,

Λ ∂z
∂ z̄
h
∂T
∂T
+
+


+
+ εR[HTαα + Re(Q(T11 − T22 + 2iT12))] + hX3 = 0;
Λ ∂z
∂ z̄
(9)
b) Hooke’s law
1
∂
u+ − εQRu3 ,
h(T11 − T22 + 2iT12) = 4µΛ h
∂ z̄ Λ
hTαα = h(T11 + T22) = 2(λ + µ)(hθ − 2HεRu3 ),
∂u3
+ εR(Hu+ + Qu+ )],
hT+ = µ(2h
∂ z̄
1 ∂u+ ∂u+
+
;
hT33 = λ(Rθ − 2HεRu3 ), θ =
Λ ∂z
∂ z̄
(10)
c) equilibrium equations in terms of the displacement vector components

1 ∂u+
4µh ∂ΛQu3
2 ∂
2 ∂θ
+
2(λ
+
µ)h
−
εR
4µh

∂ z̄ Λ ∂z
∂ z̄
∂z

Λ

 +4(λ + µ)h ∂Hu3 + 2µh H ∂u3 + Q ∂u3

∂ z̄
∂ z̄
∂z

 +ε2R2 µ (2K − H 2 − QQ)u+ − 2HQu+ + h2X+ = 0,
(11)




1
 µh2 ∇2u3 + εR 2µhRe 1 ∂(Hu+ + Qu+ ) + 2Q ∂
u+

Λ
∂z
∂ z̄ Λ
+ 2(λ + 2µ)hHθ} − 4R2 ε2 (λ + µ)H 2 + µQQ u3 + h2 X3 = 0.
To determine the components of the displacement vector and the stress
tensor we shall use expansions with respect to the small parameter ε:
(0)
(1)
(2)
ui = u i + ε u i + ε2 u i + ...
(0)
(1)
(2)
Tij = T ij + ε T ij + ε2 T ij + ...
(0)
(1)
(2)
Xi = X i + εX i + ε2 X i + ...
6
Bulletin of TICMI
and then we equate to zero the factors of εn . These equations may be written
in the form:
a)

(n)
(n)
h ∂ (n)
∂ (n) α


−
+
2i
+
h
T
T
T
T α
11
22
12


Λ ∂z
∂ z̄




(n−1)
(n−1)
(n)



=
R(H
+
Q
)
−
h
T
T
X +,
+
+



 

(12)
(n)
(n)

(n−1)
∂
∂
h

T
T


+
+
α


+

 = −R H T α


Λ
∂z
∂ z̄






(n−1)
(n−1)
(n−1)
(n)

1
2
1

 +Re Q( T 1 − T 2 + 2i T 2
− h X 3 = 0;
b) Hooke’s law
 (n)
(n)
(n)
(n−1)
∂ 1 (n)


u + − QR u 3 ,
h T 11 − T 22 + 2i T 12 = 4µΛ h


∂z̄ Λ



(n)
(n)

(n−1)

α

=
2(λ
+
µ)
h
h
T
θ − 2HR u 3 ,

α








(n)
(n)


(n)
u
u
1 ∂ + ∂ + 
 
+
θ = 
Λ
∂z
∂ z̄






(n)


(n)
(n−1)
(n−1)
∂u3



+ R(H u + + Q u + ) ,
h T + = µ 2h


∂ z̄





(n)
(n)
(n−1)


 h T 33 = λ h θ − 2HR u 3 ;
(13)
c) equilibrium equations in terms of the displacement vector components:



(n)
(n)


(n)
u
1
∂
∂
∂

+
 4µh2 
 + 2(λ + µ)h2 θ = A + (z, z̄),
∂ z̄ Λ ∂z
∂ z̄
(14)



(n)
(n)

µh2 ∇2 u 3 = A 3 (z, z̄),
where

(n)

(n)
2
A + = −h X + − R 2µh 
(n−1)
∂H u
+4(λ + µ)h
∂ z̄

3
2
− µR
2
Λ
!
(n−1)
∂ΛQ u 3
∂z
+H
(n−1)
∂ u 3
∂ z̄
" (n−2)
2K − H − QQ u
2

+Q
(n−1)
∂ u 3
∂z
(n−2)
+
− 2HQ u
+
,
Volume 8, 2004
7



(n−1)
(n−1)

(n)
(n)
∂
1 (n−1) 
1 ∂(H u + + Q u + )
2
u +
+ 2Q
A 3 = −h X 3 − R 2µhRe 

Λ
∂z
∂ z̄ Λ
(n−1)
+ 2(λ + µ)hH θ
(n−2)
+ 4R2 (λ + µ)H 2 + 2µQQ u 3 ;
d) for the boundary conditions (7), we obtain

2 
1
dz̄


T(ll) + iT(ls) =
Tαα − (T11 − T22 + 2iT12)
= f1 + if2,



2
ds

dz̄
⇒
= f3 ,
T(ln) = −Im T+



ds


0


(f i = fi ),
 2 (n)

(n)
λ
+
µ
h (n)
∂
1
dz̄


u
h
=
−Λ
f

θ
+


2µ
∂ z̄ Λ
ds
2µ +



2 

(n−1)
λ
+
µ
dz̄

 +R
u 3,
H − ΛQ
µ
ds
(15)




(n)

(n)

(n−1)
(n−1)
u
dz̄


 ∂ 3 dz̄  = − h f 3 − RIm H u + + Q u +
2hIm
,



∂ z̄ ds
µ
ds




(f+ = f1 + if2).
We may now write the general solution of the system of the equations (14)
in an explicit form [2]
(n)
u+
1
π
= −æ
S
(n)

(n)
ϕ (ζ) dS  1
+
π
ζ − z̄
S

(n)
dS  (n)
ϕ (z) − ψ (z)
ζ − z̄
B + (ζ, ζ)dS
,
ζ − z̄
S
(n)
2
u 3 = f(z) + f(z) +
B3 (ζ, ζ) ln |ζ − z|dS,
π
+
(16)
S
(ζ = ξ + iη ∈ G, dS = Λ(ζ, ζ)dξdη)
where
(n)
B + (z, z) = −
λ + 3µ
1
8µ(λ + 2µ)h2 π
G

(n)
(n)

A + (ζ, ζ) 
 A + (ζ, ζ)
−
 dξdη,
æ
ζ −z
ζ −z
8
Bulletin of TICMI
1 (n)
B 3 (z, z) =
A 3 (z, z),
4µh2
(n)
λ + 3µ
æ=
λ+µ
.
In this way the general solution of the system (14) is expressed by three
arbitrary analytic functions f(z), ϕ(z) and ψ(z) of z. Accordingly, it ensures
the satisfication of three arbitrary given physical or kinematic conditions.
4. Approximation of Order N = 1
Consider, now, the system of equations (5), (6) and (8) for constructing
approximation of order N = 1. In this case we have
x (1)
x (1)
0
(0)
3
3
σ ij ,
Ui = U i + P1
U i , σij = σ ij + P1
h
h
x (1) (0) (1) (1) (0)
3
Fi = F i + P1
F i , U i = U i , U i = 0.
h
By introducing the following notation
(0)
(1)
(0)
(1)
(0)
(1)
U i = ui, U i = vi, σ ij = Tij , σ ij = Sij , F i = Xi , F i = Yi ,
we obtain:
a) equilibrium equations:

h ∂
∂ α


(T
Tα − εR(HT+ + QT+ ) + hX+ = 0,
11 − T22 + 2iT12 ) + h


Λ
∂z
∂
z̄


 h ∂T+ ∂T+


+
+ εR HTαα + Re(Q(T11 − T22 + 2iT21)) + hX3 = 0,

Λ ∂z
∂ z̄
(17)
∂ α
∂
h


(S
S
−
S
+
2iS
)
+
h
−
εR(HS
+
QS
)
−
3T
+
hY
=
0,

11
22
12
+
+
+
+

Λ ∂z
∂ z̄ α



h ∂S+ ∂S+



+
+ εR HSαα + Re(Q(S11 − S22 + 2iS21)) − 3T33 + hY3 = 0;
Λ ∂z
∂ z̄
b) Hooke’s law
 
1
∂



 h(T11 − T22 + 2iT12) = 4µΛ h ∂ z̄ Λ u+ − εRQu3 ,





 hTαα = h(T11 + T22) = 2(λ + µ)(hθ − 2HεRu3 ) + 2λv3 ,







∂u3





 hT+ = µ 2h ∂ z̄ + εR(Hu+ + Qu+ ) + v+ ,



+ 2µ)v
3,  hT33 = λ(hθ − 2HεRu3 ) + (λ
∂
1


v+ − εRQv3 ,
h(S11 − S22 + 2iS12) = 4µΛ h


 
∂ z̄ Λ




 hSαα = h(S11 + S22 ) = 2(λ + µ)(hρ − 2HεRv3 ),






∂v

3

 

 hS+ = h(S13 + iS23) = µ 2h ∂ z̄ + εR(Hv+ + Qv+ ) ,



hS33 = λ(hρ − 2HεRv3 ),
(18)
Volume 8, 2004
where
1
θ=
Λ
∂u+ ∂u+
+
∂z
∂ z̄
1
, ρ=
Λ
∂v+ ∂v+
+
∂z
∂ z̄
9
, (u+ = u1 +iu2, v+ = v1 +iv2).
c) equilibrium equations in terms of the displacement vector components:

1 ∂u+
∂v3
4µh ∂ΛQu3
2 ∂
2 ∂θ
 4µh ∂ z̄ Λ ∂z + 2(λ + µ)h ∂ z̄ + 2λh ∂z − εR R
∂z


∂u
∂u
∂Hu
3
3
3
 + 4(λ + µ)h
+ 2µh H
+Q
+ µ(Hv+ + Qv+)

∂ z̄
∂ z̄
∂z

 +ε2R2 µ (2K − H 2 − QQ)u+ − 2HQu+ + h2 X+ = 0,


(19)


∂(Hu
+
Qu
)
1
+
+
 µ(h2 ∇2u3 + hρ) + εR 2µhRe

Λ
∂z


1
∂
2
 + 4Q
+ 2(λ + µ)h Hθ + 2λhv3
u+

∂ z̄ Λ
−4R2 ε2 (λ + µ)H 2 + 2µQQ u3 + h2 X3 = 0,

1 ∂v+
∂v3
2 ∂
2 ∂ρ
 4µh ∂ z̄ Λ ∂z + 2(λ + µ)h ∂ z̄ − 3µ 2h ∂z + v+


 −εR 4µh ∂ΛQv3 + 4(λ + µ)h ∂Hv3 + 2µh H ∂v3 + Q ∂v3

Λ
∂z
∂z
∂ z̄ 
∂ z̄
 +3µ(Hu+ + Qu+ )] + ε2R2 µ (2K − H 2 − QQ)v+ − 2HQv+

 +h2 Y+ = 0,

(20)


 µh2 ∇2v3 − 3(λhθ + (λ + 2µ)v3 )


∂
∂(Hv
1
+
Qv
)
1
+
+
 +εR 2µhRe
+ 4Q
v+

Λ
∂z#
∂ z̄ Λ

 + 2(λ + µ)h2 Hρ + 6λHv3 − 4R2 ε2 (λ + µ)H 2 + 2µQQ v3
+h2 Y3 = 0.
To determine the components of the displacement vector and the stress
tensor we shall use expansion with respect to the small parameter ε:
(0)
(0)
(1)
(1)
(2)
(2)
(ui , vi ) = ( u i , v i ) + ε( u i , v i ) + ε2 ( u i , v i ) + ...,
(0)
(0)
(1)
(1)
(2)
(2)
(Tij , Sij ) = ( T ij , S ij ) + ε( T ij , S ij ) + ε2 ( T ij , S ij ) + ...,
(0)
(0)
(1)
(1)
(2)
(2)
(Xi , Yi ) = (X i , Y i ) + ε(X i , Y i ) + ε2(X i , Y i ) + ...
and then we equate to zero the factors of εn . These equations may be written
as:
10
Bulletin of TICMI
a) equilibrium equations:

(n)
(n)
h ∂ (n)
∂ (n) α


(
−
+
2i
)
+
h

T
T
T
T α
11
22
12


Λ ∂z
∂ z̄


(n)
(n−1)
(n−1)



=
−h
+
R(H
+
Q

X
T
T + ),
+
+









(n)
(n)


(n)

h ∂ T + ∂ T +


=
−h
+



X3


Λ
∂z
∂ z̄





(n−1)
(n−1)
(n−1)
(n−1)


α
1
2
1

−R H T α + Re(Q( T 1 − T 2 + 2i T 2 )) ,








































(21)
(n)
(n)
∂ (n)
h ∂ (n)
( S 11 − S 22 + 2i S 12 ) + h
S
Λ ∂ z̄
∂ z̄
(n)
= −h Y

+
(n)
(n−1)
+ R(H S
(n)
α
α
(n)
− 3T +
(n−1)
+
+Q S
+ ),

(n)
h ∂ S + ∂ S +
+

 − 3 T 33
Λ
∂z
∂ z̄
(n)
(n−1)
(n−1)
(n−1)
(n−1)
α
1
2
1
= −h Y 3 − R H S α + Re(Q( S 1 − S 2 + 2i S 2 )) ;
b) Hooke’s law
 
(n)
(n)
(n)
(n−1)
∂ 1 (n)



 h( T 11 − T 22 + 2i T 12 ) = 4µΛ h ∂ z̄ Λ u + − RQ u 3 ,





(n)
 
(n−1)
(n)

 hT α = 2(λ + µ)(h θ − 2HR u 3 ) + 2λ v 3 ,


α







(n)

 (n)

(n−1)
(n)

 h T + = µ 2h ∂ u 3 + R(H (n−1)

u + + Q u + ) + v + ,




∂ z̄






(n)
(n)

(n−1)
(n)

u
v 3, h
−
2HR
=
λ(h
)
+
(λ
+
2µ)
T
θ
33
3

(n)
(n)
(n)
(n)
(n−1)
∂
1




 h( S 11 − S 22 + 2i u 12 ) = 4µΛ h ∂ z̄ Λ v + − RQ u 3 ,





 (n) α
(n)
(n−1)


 h S α = 2(λ + µ)(h ρ − 2HR v 3 ),







(n)
 

 (n)
(n−1)


 h S + = µ 2h ∂ v 3 + R(H (n−1)
v + + Q v + ) ,




∂ z̄







(n)
(n)
(n−1)

h S 33 = λ(h ρ − 2HR v 3 );
(22)
Volume 8, 2004
11
c) equilibrium equations in terms of the displacement vector components:



(n)
(n)
(n)


u
v 3 (n)
∂
∂
∂
1
∂

θ
+
2
2
 4µh

 + 2(λ + µ)h
+ 2λh
= L +,
∂ z̄ Λ ∂z
∂ z̄
∂z
(23)



(n)
(n)
(n)
(n)

µh2 ∇2 v 3 − 3(λh θ + (λ + 2µ) v 3 ) = M 3 ,





(n)
(n)
(n)

(n)

ρ

 4µh2 ∂  1 ∂ v +  + 2(λ + µ)h2 ∂ − 3µ 2h ∂ u 3 + (n)
v + = M +,
∂ z̄ Λ ∂z
∂ z̄
∂z
(24)


(n)

(n)
(n)

µh(h∇2 u 3 + ρ ) = L 3 ,
where

(n)
(n)
(n)
(n−1)
∂H u 3
4µh ∂ΛQ u 3
+ 4(λ + µ)h
L + = −h X + + R 
Λ
∂z
∂ z̄



(n−1)
(n−1)
(n−1)
(n−1)
∂ u 3
∂ u 3
+Q
+ µ(H v + + Q v + )
+ 2µh H
∂ z̄
∂z
(n−2)
(n−2)
2
2
−µR (2K − H − QQ) u + − 2HQ u + ,
2

(n)
(n)
(n−1)
(n−1)
∂H v 3
4µh ∂ΛQ v 3
+ 4(λ + µ)h
M + = −h Y + + R 
Λ
∂z
∂ z̄



(n−1)
(n−1)
(n−1)
(n−1)
∂ v 3
∂ v 3
+Q
+ 3µ(H u + + Q u + )
+ 2µh H
∂ z̄
∂z
(n−2)
(n−2)
2
2
−µR (2K − H − QQ) v + − 2HQ v + ,
(n)
(n)
2



(n−1)
(n−1)

1 (n−1) 
u +
Λ
∂
1 ∂(H u + + Q u + )
+ 4Q

Λ
∂z
∂ z̄
(n−1)
(n−2)
(n−1)
+2(λ + µ)hH θ + 2λ v 3 + 4R2 ε2 (λ + µ)H 2 + 2µQQ u 3 ,
2
L 3 = −h X 3 − R
2µhRe 



(n−1)
(n−1)

∂
1 (n−1) 
1 ∂(H v + + Q v + )
2
v +
+ 4Q
M 3 = −h Y 3 − R 2µhRe 

Λ
∂z
∂ z̄ Λ
(n−1)
(n−2)
(n−1)
+ 2(λ + µ)hH ρ + 6λH v 3 + 4R2 (λ + µ)H 2 + 2µQQ v 3 ;
(n)
(n)
12
Bulletin of TICMI
d) for the boundary conditions (7), we have:

2 (0)
(0)

1
dz̄

α

T
+
iT
=
−
(T
−
T
+
2iT
)
f
+
i
f 2,
T
=
(ll)
(ls)
11
22
12

1
α

2
ds



(0)


dz̄


= f 3,
 T(ln) = −Im T+
ds
⇒
2 (1)
(1)

1
dz̄


S(ll) + iS(ls) =
= f 1 + i f 2,
Sαα − (S11 − S22 + 2iS12)



2
ds


(1)



dz̄

 S(ln) = −Im S+
= f 3,
ds

2
λ + µ (n) λ (n)
∂ 1 (n)
dz̄


v 3 − hΛ
u+
hθ +



2µ
2µ ∂ z̄ Λ
ds


2 
(0,n)

(n−1)
h
λ+µ
dz̄


u ,
=
f ++R
H − ΛQ



2µ
µ
ds










(n)

(1,n)

(n−1)
(n−1)
v
dz̄
h
dz̄
∂

3

 = − f 3 − RIm H v + + Q v +
,
2hIm 



∂ z̄ ds
µ
ds






 2 ∂ 1 (n)
dz̄
λ + µ (n)
ρ −Λ
v+
h



2µ
∂ z̄ Λ
ds



2 
(1,n)

(n−1)
λ
+
µ
dz̄
h


v 3,
H − ΛQ
=
f ++R



2µ
µ
ds






 



(n)


(n)
u


2h ∂ 3 + v +  dz̄ 

 Im

∂ z̄
ds





(n−1)
(n−1)

h (0,n)
dz̄

 = − f 3 − RIm H u + + Q u +
.
µ
ds
The general solution of the homogeneous systems (23) and (24) can be written
in an explicit form [1],[2]:
u+
λh
∂ω 5λ + 6µ 1
ϕ (ζ)dS
= −
−
6(λ + 2µ) ∂ z̄
3λ + 2µ π
ζ − z̄
s


dS
1
 ϕ (z) − ψ(z),
+ 
π
ζ − z̄
s
v3 = ω −
2λh
[ϕ (z) + ϕ (z)],
3λ + 2µ
Volume 8, 2004
v+ = i
∂χ
1
− 2hΨ (z) −
∂ z̄
π
s
2
13

Φ (ζ)dS  1
−
π
ζ − z̄
s

dS  Φ (z)
ζ − z̄
4(λ + 2µ)h Φ (z),
3µ
11
= Ψ(z) + Ψ(z) −
Φ (ζ) + Φ (ζ) ln |ζ − z|ds
hπ
+
u3
s
(dS = Λ(ζ, ζ)dξη, ζ = ξ + iη)
where ϕ(z), ψ(z), Φ(z) and Ψ(z) are analytic functions of z = x1 + ix2, and
ω, χ are general solutions of the following homogeneous equations
∇2 ω −
3(λ + µ)
ω = 0,
(λ + 2µ)h2
3
χ = 0,
h2
4 ∂2
).
(∇2 =
Λ ∂z∂ z̄
∇2 χ −
References
[1] Vekua, I.N., Some General Methods for Constructing different Versions of Shell
Theory. Boston-London-Melburne, 1986.
[2] Vekua, I.N., On Construction of Approximate Solutions of Equations of the Shallow
Spherical Shell. Int. J. Solids Structures, 5 (1969), 991-1003.
[3] Meunargia, T.V., The Method of a Small Parameter for Non-Linear Spherical Shells.
Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics,
vol. 14, N2, 1999.
Received December, 12, 2003; revised May, 21, 2004; accepted June, 6, 2004.