38
Bulletin of TICMI
AN EXTENSION OF THE VEKUA-BITSADZE METHOD FOR SOLVING
EQUATIONS OF THE SHELLS
Meunargia T.
I. Vekua Institute of Applied Mathematics of
Iv. Javakhishvili Tbilisi State University
Key words: Shell, Vekua-Bitsadze method parameter, nonlinear theory.
MSC 2000 : 74K25.
I. Vekua has constructed several versions of the refined linear theory of thin
and shallow shells, containing the regular process by means of the method of
reduction of three-dimensional problems of elasticity to two-dimensional ones.
In the present paper by means of the I. Vekua method the system of differential equations for the nonlinear theory of non-shallow shells is obtained.
Using the method I. Vekua and the method of a small parameter 2-D system
of equations for the nonlinear and non-shallow shells is obtained. For any approximations of order N the complex representations of Vekua-Bitsadze type
[2] of the general solutions are obtained.
Under thin and shallow shells I. Vekua meant three-dimensional shell-type
elastic bodies satisfying the conditions
aβα − x3 bβα ∼
= aβα , −h(x1 , x2 ) ≤ x3 ≤ h(x1 , x2 ) (α, β = 1, 2),
(∗)
where aβα and bβα are mixed components of the metric tensor and the curvature
tensor of the shell’s midsurface, x3 is the thickness coordinate and h is the
semi-thickness, depending on curvilinear coordinates x1 , x2 .
In the sequel, under by non-shallow shells we mean elastic bodies not subject to assumption (*), i. e., such that
¯
¯
aβα − x3 bβα aβα ⇒ ¯x3 bβα ¯ ≤ q < 1.
1. To construct the theory of shells we use the coordinate system which is
normally connected with the midsurface S. This means that the radius-vector
of any point of the domain Ω can be represented in the form [1]
R(x1 , x2 , x3 ) = r(x1 , x2 ) + x3 n(x1 , x2 ) (x3 = x3 ),
where r and n are radius-vector and the unit vector of the normal of the
midsurface S(x3 = 0); x1 , x2 are the Gaussian parameters of S.
Covariant and contravariant basis vectors Ri and Ri of the surface
Ŝ(x3 = const) and the corresponding basis vectors r i and r i of the midsurface
S(x3 = 0) are connected by the following relations [1]:
Ri = A.ji. r j = Aij r j , Ri = Ai..j r j = Aij r j (i, j = 1, 2, 3),
Volume 13, 2009
39
where
β
β
α.
−1
α
α
i
3
3
A.β
α. = aα − x3 bα , A.β = ϑ [(1 − 2Hx3 )aβ + x3 bβ ], A3 = Ai = δi ,
ϑ = 1 − 2Hx3 + Kx23 , R3 = R3 = r 3 = r 3 = n (α, β = 1, 2).
(1)
Here (aαβ , aαβ , aαβ ) and (bαβ , bαβ , bαβ ) are the components (co, contra, mixed)
of the metric tensor and curvature tensor of the midsurface S. By H and K
we denote a middle and Gaussian curvature of the surface S, where
2H = bαα = b11 + b22 , K = b11 b22 − b12 b21 .
The main quadratic forms of the midsurface S have the form
I = ds2 = aαβ dxα dxβ ,
II = ks ds2 = bαβ dxα dxβ ,
(2)
where ks is the normal curvature of the surface S, and
aαβ = r α r β , bαβ = −r α nβ , ks = bαβ sα sβ , sα =
dxα
.
ds
Here and in the sequel, under a repeated indices we mean summation; note
that the Greek indices range over 1, 2, while Latin indices range over 1, 2, 3.
To construct the theory of non- shallow shells, it is necessary to obtain formulas for a family of surfaces Ŝ(x3 = const), analogous to (2) of the midsurface
S(x3 = 0) which have the form [1]
I = dŝ2 = gαβ dxα dxβ ,
II = kŝ dŝ2 = b̂αβ dxα dxβ ,
(3)
where
gαβ = aαβ − 2x3 bαβ + x23 (2Hbαβ − Kaαβ ), b̂αβ = (1 − 2Hx3 )bαβ + x3 Kaαβ ,
and kŝ the normal curvature of the surface Ŝ.
It is not now difficult to get the expression for the tangential normal l̂ of
the surface Ŝ directed to ŝ [3]:
l̂ = ŝ × n = [(1 − x3 ks )l − x3 τs s]
ds
, dŝ =
dŝ
q
1 − 2x3 ks + x23 (ks2 + τs2 )ds,
where s and l are the unit vectors of the tangent and tangential normal on
S, dŝ and ds are the linear elements of the surfaces Ŝ and S, and τs is the
geodesic torsion of the surface S.
2. We write the equation of equilibrium of elastic shell-type bodies in a
vector form
√
1 ∂ gσ i
+ Φ = 0, ⇒ ∇i σ i + Φ = 0,
(4)
√
g ∂xi
40
Bulletin of TICMI
where g is the discriminant of the metric quadratic form of the three-dimensional
domain Ω, ∇i are covariant derivatives with respect to the space coordinates
xi , Φ is on external force, σ i are the contravariant constituents of the stress
∗
vector σ ∗ acting on the area with the normal l and representable as the
(l)
Cauchy formula as follows:
µ
¶
∗
∗
∗
i
σ ∗ = σ l i , l i = l Ri ,
.
(l)
For the stress vector acting on the area with normal l̂, we obtain
σ (l̂) = σ α (l̂Rα ) = ϑσ α (lr α )
ds
.
dŝ
(5)
The stress-strain relation for the geometrically nonlinear theory of elasticity
has the form
σ i = σ ij (Rj + ∂j U ) = E ijpq epq (Rj + ∂j U ),
(6)
where σ ij are contravariant components of the stress tensor, eij are covariant
components of the strain tensor, U is the displacement vector, E ijpq and eij
are defined by the formulas:
1
E ijpq = λg ij g pq + µ(g ip g jq + g iq g ip ), eij = (Ri ∂j U + Rj ∂i U + ∂i U ∂j U ). (7)
2
To reduce the three-dimensional problems of the theory of elasticity to the
two-dimensional problems, it is necessary to rewrite the relation (4-7) in forms
of the bases of the midsurface S of the shell Ω.
The relation (4) can be written as
√
1 ∂ aϑσ α ∂ϑσ 3
√
+
+ ϑΦ = 0, (a = a11 a22 − a212 ).
(8)
∂x3
a ∂xα
From (1), (6), (7) we obtain
1
σ i = Aii1 App1 , M i1 j1 p1 q1 [(r q1 ∂p U ) + Aqq1 (∂p U ∂q U )](r j1 + Ajj1 ∂j U ),
2
(9)
M i1 j1 p1 q1 = λai1 j1 ap1 q1 + µ(ai1 p1 aj1 q1 + ai1 q1 aj1 p1 ) (aij = r i r j )
3. The isometric system of coordinates on the surface S is of special interest, for in this system we can obtain basic equations of the theory of shells in
a complex form which in turn allows one to construct for a rather wide class
of problems complex representations of general solutions by means of analytic
functions of one variable z = x1 + ix2 .
The main quadratic forms in the system of coordinates are of the type
I = ds2 = Λ(z, z̄)dzdz̄,
¤
1 £
II = ks ds2 = Λ Q̄dz 2 + 2Hdzdz̄ + Qdz̄ 2 ,
2
Volume 13, 2009
Q = 0.5(b11 − b22 + 2ib12 ),
41
Λ(z, z̄) > 0.
Introducing the well-known differential operators 2∂z = ∂1 − i∂2 , 2∂z̄ =
∂1 + i∂2 and the notations τ.ji. = ϑσ i r j , Xi = ϑΦr i , for the geometrically
nonlinear theory of non-shallow shells from (8) and (9) we obtain the following
complex form both for the system of equations of equilibrium and Hooke’s law:
3
∂τ·+
1 ∂Λτ + r + ∂ τ̄ + r +
+
·+
+ X+ = 0,
+
− Λ(Hτ3 + Qτ̄3 ) +
Λ µ ∂z
∂ z̄ ¶
∂x3
+
+
¡
¢ ∂τ 3
1 ∂Λτ3
∂Λτ̄3
+ H(τ11 + τ22 ) + Re Q̄τ + r + + 3 + X3 = 0,
+
Λ
∂z
∂ z̄
∂x3
©
£
¡
¢
¤ª ¡ +
¢
τ + = ϑ λΘ+µ R+ ∂z U + R̄+ +2∂z U ∂ z̄ U
R +2∂ z̄ U
n£
´
¤³ +
+
z̄
z
+ µϑ R ∂z̄ U +(R+ +2∂z̄ U ) ∂ U R̄ +2∂ U
£
¤
ª
+ R+ ∂3 U +(n+∂3 U ) ∂ z̄ U (n+∂3 U) ,
¸
½·
1
3
3
3
τ = ϑ λΘ + 2µ(n∂ U + ∂3 U ∂ U ) (n + ∂3 U )
2
·
1
+µ ( R̄+ ∂3 U + n∂z U + ∂z U ∂3 U )(R+ + 2∂ z̄ U )
2
¸¾
1
+
z
+ ( R+ ∂3 U + n∂z̄ U + ∂z̄ U ∂3 U )(R̄ + 2∂ U ) ,
2
(10)
(11)
(∂ 3 U = ∂3 U ).
Here
¡
¢
¡
¢
τ + r + = τ 1 + iτ 2 (r 1 + ir 2), τ̄ + r + = τ 1 − iτ³2 (r 1 + ´
ir 2), τ.3+. = τ + n,
¡
¢
+
3.
τ.+
= τ 3 r + , τ33 = τ 3 n, 2∂ z̄ U = R+ R+ ∂z U + R+ R̄ ∂z̄ U, τ.βα. = τ α r β ,
£¡
¤¢
£
Θ = 2Re ¤ R+ +∂ z̄ U ∂z U +∂3 U 3 +0.5 (∂3 U )2 , R+ = ϑ−1 (1−Hx3 ) r ++
+x3 Qr̄ + , R+ = (1 − Hx3 )r + −x3 Qr̄ + , R+ = R1 +iR2, R+ = R1 +iR2,
+
R+ R+ = 4x3 (Λϑ2 )−1 (1−Hx3 )Q, R+ R̄ = 2(Λϑ2 )−1 (ϑ+2x23 QQ̄),
+
R+ r + = 2ϑ−1 Qx3 , R̄ r + = 2ϑ−1 (1 − Hx3 ), r + r̄ + = 2Λ−1 , r + = r 1 + ir 2 .
We have the formulas
+
+
r + ∂z U = Λ−1 ∂z U+ −
¡ HU3 , r ∂z̄¢U =+∂z̄ U −+QU3 , X+ = X1 +iX2 ,
n∂z U = ∂z U3 + 0.5 Q̄U+ +H Ū+ (U = U r , U+ = U r + , U3 = U n) .
4. In the present paper the three-dimensional problems of the theory of
elasticity are reduced to the two-dimensional ones by the
n method
³ x ´osuggested by
3
is complete
I. Vekua. Since the system of Legendre polynomials Pm
h
42
Bulletin of TICMI
in the interval [-h,h], for equation (8) we obtain the infinite system of twodimensional equations
Zh ·
¸
√
¡ ¢
1 ∂ aϑσ α ∂ϑσ 3
√
+
+ ϑΦ Pm xh̄3 dx3 = 0 (m = 0, 1, ...)
α
3
∂x
a ∂x
−h
or in a form
(m) α
∇α σ
2m + 1
−
h
µ
(m−1) 3
σ
¶ (m)
+ ... + F = 0,
(m−3) 3
+ σ
(12)
where
µ
(m) i (m)
¶
σ ,Φ
2m + 1
=
2h
Zh
¡ i
¢ ¡ ¢
2m + 1
ϑσ , ϑΦ Pm xh̄3 dx3 =
2h
−h
(m)
(m)
2m + 1
F = Φ+
2h
Zh
¡
¢
τ i , X Pmdx3 ,
−h
µ
(+) (+)
3
m
ϑ σ −(−1)
(−) (−)
ϑ σ
¶ µ
¶
ϑ = ϑ(±h) ,
(±)
3
∇α are covariant derivatives on the midsurface S.
The equation of state (9) may be written as
"Ã
!µ
¶
∞
∞
(m2 )
(m1 )
X
X
(m)
(m)
(m) i
ip
ijp
i1 j1 p1 q1
σ =M
r q1 ·Dp U
A i1 p1r j1 +
A i1 j1 p1Dj U
(m1 )
(m1 ,m2 )
m
=0
m
=0
1
2
Ã
!µ
¶#
∞
∞
(m2 )
(m1 )
(m3 )
X
(m)
(m)
1 X
ipq
ij pq
rj +
Dj U
Dp U Dq U
+
A
A
2 m =0 (m1 ,m2i)1 p1 q1 1 m =0 (m1 ,m2 ,m3i)1 j1 p1 q1
2
(13)
3
where
(m)
Di U =
δiβ ∂β
(m)
A
(m1 )
ij
i1 j1
(m)
U
+δi3
2m + 1
=
2h
(m)
ijpq
A i1 j1 p1 q1
(m1 ,m2 ,m3 )
0
0
Zh
ϑAii1 , Ajj1 , Pm1
µ(m+1) (m+3)
¶
U + U +... ,
³x ´
3
h
Pm
³x ´
3
h
(14)
dx3
−h
ijp
A i1 j1 p1
(m1 ,m2 )
(m)
(m)
2m + 1
U ; U =
h
(m)
2m + 1
=
2h
2m + 1
=
2h
Zh
ϑAii1 , Ajj1 , App1 Pm1 Pm2 Pm dx3 ,
−h
Zh
ϑAii1 , Ajj1 , App1 Aqq1 Pm1 Pm2 Pm3 Pm dx3 .
−h
The boundary conditions on the lateral contour ∂S take the form:
(15)
Volume 13, 2009
43
a) for the stresses
σ
2m + 1
= σ (ll) l + σ (ls) s + σ (ln) n =
2h
(m)
(m)
(l)
(m)
(m)
Zh
σ (l)
³x ´
dŝ
3
Pm
dx3 ,
ds
h
(16)
h
b) for the displacements
2m + 1
U=
2h
m
Zh
U Pm
³x ´
3
h
(m)
dx3 = U
(m)
(l) l + U
(m)
(s) (s) + U 3 n.
(17)
h
Thus we have constructed an infinite system of two-dimensional equations
of geometrically non-linear and non-shallow shells (12-17), which is consistent
(±)
with the boundary conditions on the face surfaces, i.e. σ 3 = σ 3 (x1 , x2 , ±h).
The passage to finite systems can be realized by various methods one of
which consists in considering of a finite series, i.e.
¶
N µ
³x ´
X
(m) i (m) (m)
3
i
σ , U , Φ Pm
(ϑσ , U , ϑΦ) =
= (τ i , U , X),
h
m=0
where N is a fixed nonnegative number. In other words, it is assumed that
(m)
(m)
U = 0, σ i = 0, if m > N.
Approximation of this type will be called approximation of order N .
The integrals of type (15) can be calculated [5], for example,
(m)
A
(m1 )
αβ
α1 β1
2m + 1
=
2h
Zh
ϑ
−1
Bαα1 (x3 )Bββ1 (x3 )Pm1
³x ´
3
h
Pm
³x ´
3
h
dx3 =
−h
·
µ
¶¸y2 α β
Lα Lβ m
2m + 1 α
Pm1 (y)Qm (y), m1 ≤ m
β
√
Bα1 (hy)Bβ1 (hy)
+ 1 1 σm
,
1
Qm1 (y)Pm (y), m1 ≤ m y1
K
2 Eh
(18)
m
, if E = H 2 − K = 0; where Qm (y) is
if E 6= 0 K 6= 0 and aαα1 aββ1 δm
1
the Legendre function of the second kind, E is the Euler difference, Bβα (x) =
aαβ + xLαβ , Lαβ = bαβ − 2Haαβ . Under the square brackets we mean the following:
√
[f (y)]yy21 = f (y2 ) − f (y1 ), y1,2 = [(H ∓ E)h]−1 .
For the integrals containing the product of three (four) Legendre polynomials we have
(m)
A
(m1 ,m2 )
α1 α2 α3
β1 β2 β3
2m + 1
=
2n
Zh
−h
Bβα11 Bβα2 Bβα3
2m + 1
Pm1 Pm2 Pm dx3 =
×
1 − 2Hx3 + Kx3
K 2 h4
44
Bulletin of TICMI
"
Ã
!#y2
2
n
P
(y)Q
(y),
s
≤
m
s
m
y
∂
α α α n
αm1 m2 r
,
Cβ11β22β33 h
∂y
∂y
y
−
y
Q
(y)P
(y),
s
≥
m
1
2
1
2
s
m
n=0
y
min(m1 ,m2 )
×
X
r=0
3
X
n
1
where s = m1 + m2 − 2r,
apqr =
Ap−r Ar Aq−r 2(p + q) − 4r + 1
1.3 · · · 2p − 1
, Ap =
,
Ap+q−r 2(p + q) − 2r + 1
p!
Bβα11 (x)Bβα22 (x)Bβα33 (x)
=
3
X
n
α α α n
Cβ11β22β33 x ,
n=0
5. Three-dimensional shell-type bodies are characterized by inequalities
|hbαβ | ≤ q < 1 (α, β = 1, 2)
Therefore they can be represented as follows
|εbαβ R| ≤ q < 1,
where ε = hR−1 is a small parameter.
Here h is semi-thickness of the shell, R is a certain characteristic radius of
curvature of the midsurface S [4].
Now, following Signorini [3] we assume the validity of the expansions
!
Ã
¶ X
µ
∞
(m,n) (m,n) (m,n)
(m) i (m) (m)
εn .
σi , U , F
σ , U, F =
n=1
Substituting the above expansions into the (12,13) and (10,11) than equalizing the coefficients of expansions for εn we obtain the following 2-D finite
system of equilibrium equations with respect to components of displacement
vector in the isometric coordinates a11 = a22 = Λ(x1 , x2 ), which has the form:
³
´
(m,n)
(m,n)
(m,n)
4µ∂z Λ−1 ∂z u+ + 2(λ + µ)∂z θ + 2λ∂z u03 − (2m + 1)µ
i (m,n)
h ³(m−1,n) (m−3,n)
´ (m−1,n) (m−3,n)
2∂z u3 + u3 + · · · + u0+ + u0+ + · · · + F+ = 0, (19)
³ (m,n) (m,n)´
h ³(m−1,n) (m−3,n)
´
µ ∇2 u3 + θ0 − (2m + 1) λ
θ + θ +··· +
´i (m,n)
³(m−1,n) (m−3,n)
(λ + 2µ) u03 + u03 + · · · + F3 = 0,
³
´
where u+ = u1 + iu2 , θ = Λ−1 ∂z u+ ∂z u+ , z = x1 + ix2 ,2∂z = ∂1 − i∂2 ,
∇2 =
4 ∂2
.
Λ ∂z∂z
Volume 13, 2009
45
Obviously, in passing from the n-th step of approximation to the (n + 1)-th
step only the right-hand side of equations are changed. Below we will omit
upper index n.
Consider now the cases: N = 0, 1, 2, 3 [6].
Case N = 0. From (3) we get
³
´
(0)
(0)
4µ∂z Λ−1 ∂z u+ + 2(λ + µ)∂z θ = 0,
³(m) (m)
´
(0)
µ∇2 u3 = 0
F+ = F3 = 0, m = 0, 1, 2, 3 .
(20)
The complex representation of general solutions has the form
ZZ 0
ZZ 0
(0)
λ + 3µ 1
ϕ (ζ)dS
1
ϕ (ζ)dS
u+ = −
+
− ψ(z),
λ+µ π
π
S ζ −z
S ζ −z
³
´
(0)
u3 = f (z) + f (z), dS = Λ(ζ, ζ)dζ dζ̄, ζ = ξ + iη ,
(21)
where f (z), ϕ(z) and ψ(z) are holomorphic functions of z. We note that
(0)
for plane (i.e. Λ = 1) the expression of u+ coincides with the well-known
representation of Kolosov-Muskhelishvili.
(0) (1)
(1) (0)
Case N = 1. With respect to the components (u+ , u3 ) and (u1 , u3 ) we
have two systems of equations:
³
−1
4µ∂z Λ
(0)
∂z u+
´
(0)
(1)
+ 2(λ + µ)∂z θ + 2λ∂z u3 = 0,
h
(0)
(1)i
(1)
µ∇2 u3 + 3 λ θ + (λ + 2µ) θ = 0
and
³
´
³ (0) (1) ´
(1)
(1)
4µ∂z Λ−1 ∂z u+ + 2(λ + µ)∂z θ − 3µ 2∂z u3 + u+ = 0,
(0)
∇ 2 u3
(1)
(22)
(23)
+ θ = 0.
The complex representation of general solutions has the form:
ZZ 0
ZZ 0
(0)
5λ + 6µ 1
ϕ (ζ)dS
1
ϕ (ζ)dS
λ
∂w
u+ = −
+
− ψ(z) −
3λ + 2µ π
π
6(λ + µ) ∂z
S ζ −z
S ζ −z
µ
¶
(0)
12(λ + µ)
2λ
(ϕ0 + ϕ0 ), ∇2 w =
w ,
(24)
u3 = w −
λ + 2µ
λ + 2µ
and
ZZ
Φ0 (ζ) + Φ0 (ζ)
4(λ + µ) 00
∂χ
dS +
Φ (z) − 2Ψ0 (z) + i ,
3µ
∂z
ζ−
S
Z Zz ³
´
(0)
u3 = Ψ(z) + Ψ(z) −
Φ0 (ζ) + Φ0 (ζ) ln |ζ − z|dS (∇2 χ = 3χ),
(1)
u+
1
=−
π
S
46
Bulletin of TICMI
where Φ(z) and Ψ(z) are holomorphic functions of z.
Note that the systems (10) and (11) coincide with I. Vekua’s refined systems
of equations for the stretch-strain and bending of plate, respectively.
(0) (2) (1)
Case N = 2. In this case with respect to the components (u1 , u+ , u3 ) and
(1) (0) (2)
(u+ , u3 , u3 ) we have two systems of equations:
³
´
(0)
(0)
(1)
4µ∂z Λ−1 ∂z u+ + 2(λ + µ)∂z θ + 2λ∂z u3 = 0,
³
´
³ (1)
´
(2)
(2)
(2)
4µ∂z Λ−1 ∂z u+ + 2(λ + µ)∂z θ − 5µ 2∂z u3 + 3u+ = 0,
³ (1)
h (0)
i
(2)´
(1)
2
µ ∇ u3 + 3 θ − 3 λ θ + (λ + 2µ)u3 = 0
and
(25)
³
´
³ (0) (1) ´
(1)
−1 (1)
4µ∂z Λ ∂z u + 2(λ + µ)∂z θ − 3µ 2∂z u3 + u+ = 0,
(0)
(1)
∇2 u3 + θ = 0,
h (1)
i
(2)
(2)
µ∇2 u3 − 5 λ θ + 3(λ + 2µ)u3 = 0.
(26)
In this case the complex representations of the general solutions of the
equations (12) and (13) take the forms
ZZ 0
ZZ 0
(0)
5λ + 6µ 1
1
ϕ (ζ)dS
ϕ (ζ)dS
u+ = −
+
− ψ(z)
3λ + 2µ π
π
S ζ −z
S ζ −z
2
2λ X 1 ∂vk
−
,
λ + 2µ k=1 αk ∂z
µ
¶
2
X
(2)
2 ∂ω
α3−k ∂vk
2λ
00
u+ =
i
+
ϕ (z) +
,
(27)
3 ∂z
3λ + 2µ
αk ∂z
k=1
¢
(1)
2λ ¡ 0
ϕ (z) + ϕ0 (z) ,
u3 = v 1 + v 2 −
λ + 2µ
where
∇2 vk = αk vk , αk2 −
and
12(λ + µ)
180µ(λ + µ)
αk +
= 0 (k = 1, 2); ∇2 ω = 15ω,
λ + 2µ
(λ + 2µ)2
ZZ
16(λ + µ) 00
Φ0 (ζ) + Φ0 (ζ)
dS +
Φ (z)−
3(λ + 2µ)
ζ −z
S
(28)
∂χ
λ
∂w
0
−
,
−2Ψ (z) + i
∂z
10(λ + µ) ∂z
ZZ ³
´
(0)
1
λ
Φ0 (ζ) + Φ0 (ζ) ln |ζ − z|dS +
w,
u3 = Ψ(z) + Ψ(z) −
π
20(λ + µ)
S
´
(2)
2λ ³ 0
u3 = w −
Φ (z) + Φ0 (z) ,
3λ + 2µ
(1)
u+
1
=−
π
Volume 13, 2009
where ∇2 w =
60(λ+µ)
λ+2µ
47
w, ∇2 χ = 3χ.
Case N = 3. For this case we have
(0)
u+
5λ + 6µ 1
=−
3λ + 2µ π
ZZ
S
1
ϕ0 dS
+
ζ −z π
ZZ
3
S
(2)
u+
µ
¶
3 (2)
X
∂v
2 ∂ω
2λ
k
i
ϕ00 +
Ak
,
=
+
3 ∂z
3λ + 2µ
∂z
k=1
(1)
u3
3 (1)
X
=
Ak vk −
(3)
u3
=
k=1
3
X
³
vk ,
(1)
2λ X 1 + Ak ∂vk
ϕ0 dS
−ψ−
λ + 2µ k=1 αk
∂z
ζ −z
(29)
2λ
(ϕ0 + ϕ0 ),
3λ + 2µ
2
2
´
∇ vk = αk vk , ∇ ω = 15ω ,
k=1
where
αk3 −
7 · 900(λ + µ)
180(λ + µ) 2 120(λ + µ)(7λ + 15µ)
αk +
αk +
= 0,
2
λ + 2µ
(λ + 2µ)
λ + 2µ
·
¸·
¸−1
3(9λ + 4µ)
180µ(λ + µ)
12(λ + µ)
180µ(λ + µ)
2
Ak =
αk −
αk −
αk +
,
λ + 2µ
(λ + 2µ)2
λ + 2µ
(λ + 2µ)2
¸·
¸−1
·
(2)
λ
12(λ + µ)
12(λ + µ)
180µ(λ + µ)
2
αk −
αk −
αk +
,
Ak = −10
λ + 2µ
λ + 2µ
λ + 2µ
(λ + 2µ)2
(1)
and
(1)
u+
(3)
u+
(0)
u3
(2)
u3
ZZ
Φ0 + Φ0
4 23λ + 24µ 00
dS +
Φ − 2Ψ +
15 λ + 2µ
S ζ −z
¶
2 µ
X
κk − 3 ∂χk
6λ ∂wk
+
i
−
,
3
∂z
λ
+
2µ
∂z
k=1
µ
2
X ∂χk 2 γ3−k ∂wk ¶
4 3λ + 2µ 00
+
−
=
i
Φ (z),
∂z
5
γ
∂z
15
λ
+
2µ
k
k=1
1
= −
π
(30)
(∇2 wk = γk wk , ∇2 χk = κk χk , ),
¶
ZZ
2 µ
X
γ3−k 1
3λ
−
wk −
(Φ0 + Φ0 ) ln |ζ − z|dS + Ψ + Ψ,
=
λ
+
2µ
5
γ
k
S
k=1
=
2
X
wk −
k=1
µ
γk2
2λ
(Φ0 + Φ0 )
3(λ + 2µ)
¶
60(λ + µ)
35µ(λ + µ)
2
−
γk + 60
= 0, κk − 45κk + 105 = 0 .
3(λ + 2µ)
(λ + 2µ)2
48
Bulletin of TICMI
The general solution of the homogeneous system (19) we can find the form
³ 1 Z Z ϕ0 (ζ) − æ ϕ0 (ζ)dS
´
1 0
ζ
0
= ∂z V + +
− ψ00 (z) δ0m −
π
ζ −z
S
´
³ 1 Z Z ϕ0 (ζ) + ϕ0 (ζ)dS
ζ
1
1
+ η1 ϕ001 (z) − 2ψ10 (z) δ1m
π
ζ −z
(m)
(m)
u+
S
(31)
+æ2 ϕ000 (z)δ2m + η2 ϕ001 (z)δ3m ,
Z
Z
³1
´
(m)
(m)
u3 = V3 −
(ϕ01 (ζ) + ϕ01 (ζ))ln|ζ − z|dSζ − ψ1 (z) − ψ1 (z) δ0m
π
S
i
3 h 0
0
0
0
(m = 0, 1, ..., N )
− æ2 (ϕ0 (z) + ϕ0 (z))δ1m − (ϕ1 (z) + ϕ1 (z))δ2m
2
(0)
(0)
(0)
V1 = V2 = 0, u3 = ψ1 (z) + ψ1 (z), if N = 0
(dSζ = Λ(ζ, ζ)dζdζ, ζ = ξ + iη).
where ϕ00 (z),ϕ01 (z),ψ00 (z),ψ10 (z) are holomorphic functions of z and express the
biharmonic solution of the system (19). Then æ1 , æ2 , η1 , η2 are known constants.
(m)
Substituting expressions (31) into (19) the matrix equations for Vi are
obtained
∇2 V − AV = X, ∇2 Ω − BΩ = Y,
(32)
where V and Ω are column-matrices of the form
(N ) (0) (1)
(N )¢
¡ (0) (1)
T
V = V1 , V1 , ..., V1 , V3 , V3 , ..., V3 ,
(N )¢
¡ (0) (1)
T
Ω = V2 , V2 , ..., V2 ,
and A and B are block-matrices 2N +2×2N +2 and N +1×N +1 respectively.
Using now the formulae Vekua-Bitsadze for the homogenous matrix equations (32) we obtain the following complex representation of the general solutions
A
V = 2Re{ϕ(z) +
4
Zz Zz
Λ(t, t)R(z, z, t, t)ϕ(t)dtdt},
z0 z 0
B
Ω = 2Re{f (z) +
4
Zz Zz
Λ(t, t)r(z, z, t, t)f (t)dtdt},
z0 z 0
or
Zz
h
i
V (z, z) = 2Re αR(z, z, z0 , z 0 )ϕ(z) + Φ(t)R(z, z, t, z 0 )dt ,
z0
Volume 13, 2009
49
Zz
h
i
Ω(z, z) = 2Re βr(z, z, z0 , z 0 ) + Ψ(t)r(z, z, t, z 0 )f (t)dt ,
z0
³
α
ϕ(z) =
2
Zz
Φ(t)dt, V (z0 , z 0 ) = α,
Φ(t) =
∂V (z0 , z 0 ) ´
,
∂z
Ψ(t)dt,
Ψ(t) =
∂Ω(z0 , z 0 ) ´
,
∂z
z0
³
β
f (z) =
2
Zz
Ω(z0 , z 0 ) = β,
z0
where R and r are the Riemann’s matrix functions of the equations (32), ϕ(z)
and f (z) are holomorphic column-matrices:
ϕ(z) = (ϕ0 (z), · · ·, ϕN (z), ϕN +1 (z), · · ·ϕ2N (z))T , f (z) = (f0 (z), · · ·, fN (z))T .
Then particular solutions of the matrix equations (32) have the form
1
V (z, z) =
4
∧
Zz Zz
Λ(t, t)R(z, z, t, t)X(t, t)dtdt,
z0 z 0
1
Ω(z, z) =
4
∧
Zz Zz
Λ(t, t)r(z, z, t, t)Y (t, t)dtdt,
z0 z 0
where
A
R(z, z, t, t) = E +
4
Zz Zz
Λ(t1 , t1 )dt1 dt1 +
t
t
³ A ´2 Zz Zz
³ Zt1 Zt1
´
Λ(t1 , t1 )
Λ(t2 , t2 )dt2 dt2 dt1 dt1 · · ·
4
t
t
t
r(z, z, t, t) = E +
B
4
Zz
t
z
Z
Λ(t1 , t) dt1 dt1 +
t
t
³ B ´2 Zz Zz
³ Zt1 Zt1
´
Λ(t1 , t1 )
Λ(t2 , t2 )dt2 dt2 dt1 dt1 + · · ·
4
t
t
t
t
For the first boundary condition (in stress) we have
³ ∂ u dz ´ (m)
(m)
∂ u+ ³ dz ´2 (m)
3
= a1 + i b1 , Im
= c1 (on ∂S)
(λ + µ) θ − 2µΛ
∂z ds
∂z ds
(m)
(m)
(m)
50
Bulletin of TICMI
The second boundary condition (in displacements) for any m takes the form
(m) dz
u+
ds
(m)
(m)
(m)
(m)
= a2 + i b2 , u3 = c2 (on ∂S).
The basic boundary conditions (N = 0) for any n have the form:
a) for the first boundary problem (in displacements)
(0,n)
(0,n)
(0,n)
U (`) + i U (s) = i U +
dz̄ (n)
= d+ ,
ds
(0,n)
U
3
(n)
= d 3 on ∂D
(33)
b) for the second boundary problem (in stresses)
·
µ
¶ ¸
(n)
(0,n)
1 (0,n) +
dz
σ (``) + i σ (`s) =
σ̄ r + − σ + r +
= e +,
dz̄
¶ ¸
·µ 2
(n)
(0,n)
(0,n)
dz̄
σ (`n) = −Im
σ +n
= e3 on ∂D.
ds
(0,n)
(0,n)
(34)
Here we present a general scheme of solution of boundary problems when
the domain D is a circle of radius r0 .
The first boundary problem for any n takes the form (on |z| = r0 ),
(0,n)
U
+
æ
=−
π


ZZ
(n)
Λϕ0 (ζ)dξdη  1
Λdξdη  0
ϕ (z) − ψ(z) = G +
+
π
ζ̄ − z̄
ζ̄ − z̄
ZZ
D
D
(0,n)
U
(n)
(35)
3
(n)
= f (z) + f (z) = G 3 (z = reiϕ , ζ = ρeiψ ),
(n)
(0,1)
(36)
(0,n−1)
where G + and G 3 are the known values containing solutions U i , · · · , U i of
the previous approximations.
(n)
Let Λ(z, z̄) depend only on r = |z|, next ϕ0 (z), ψ(z) and G + are expanded
in power series of the type
0
ϕ (z) =
∞
X
k=0
k
ak z , Ψ(z) =
∞
X
k
∞
X
(n)
bk z , G + =
k=0
Ak eikϑ .
k=−∞
Substituting these expansions into (24), we obtain
a0 =
r0k+1 Ak+1
r0 æA1 + Ā1
,
a
=
(k ≥ 1),
k
α0 æ2 − 1
æαk
Āk α0 r0k+2
bk = − k −
Ak+2 , (k ≥ 0), αk = 2
æαk+1
r0
Zr0
ρ2k+1 Λ(ρ)dρ.
0
Volume 13, 2009
(0,n)
U
3
51
is representable in the form of the Poisson integral,
1
U 3 (r, ϑ) =
2π
(0,n)
Z2π (n)
G 3 (ψ)
0
r02 − r2
dψ.
r2 − 2r0 r cos(ψ − ϑ) + r02
(37)
Thus for any n we can construct formal solutions of the problem (22), when
N = 0.
From the second boundary condition (23), we obtain (on ∂D)
µ
¶
(n)
(0,n)
(n)
(0,n)
(0,n)
1 (0,n) dz̄
σ (``) + i σ (`s) = ` + ⇒ (λ + µ) Θ −2µ
= P +,
(38)
U+
Λ
dz
 (0,n) 
(n)
(0,n)
∂ U 3 dz̄  (n)
σ (`n) = `3 ⇒ Im 
(39)
= P3 ,
∂z ds


(0,n)
(0,n)

1  ∂ U + ∂ Ū + 
(0,n)
+
Θ = 
 .
Λ
∂z
∂ z̄
Consider the case of a spherical shell, whose midsurface is a spherical segment of radius R0 sin ϑ, where R0 is the radius of a sphere. Isometric coordinates on the sphere can be represented in the form
ϑ
z = x1 + ix2 = reiϕ , r = tg , Λ = 4R2 (1 + z z̄)−2
2
(0 ≤ ϑ ≤ ϑ0 ).
Let the expressions
µ
¶
∞
∞
X
X
(n) (n)
k
g+ , g3 =
(ak , bk , ck )z ,
(Ak , Bk )eikϕ ,
(ϕ (z), Ψ (z), f (z)) =
0
0
k=0
(n)
k=−∞
(0,1)
(n)
(0,n−1)
be valid, where g+ and g3 are known values expressed by U i , · · · , U i of
the previous approximations. Substituting these expansions into (27), (28)
and taking into account that principal vector and moment of stresses are zero,
we obtain
1
Ak
,
k 1 + 2æ(1 + r 2 )β
2µr0
k
0
·
¸
−1 (1 + r02 )−1 ((1 + r02 )k + 2r02 )Ak+1
bk =
+ Āk−1
(k ≥ 0).
1 + 2æ(1 + r02 )βk+1 r02
2µr0k−1 k + 2r02
Zz
1
2 R0
Bk
(z − t)tk dt
(k
≥
1),
B
=
0,
β
(z)
=
ck =
.
0
k
µ 1 + r02 kr0k−1
z k+2
(1 + tz̄)3
ak =
0
52
Bulletin of TICMI
From here we obtain the well-known Dini’s formula
r0
U 3 (r0 , ϕ) = −
π
(0,n)
Z2π (n)
iϑ
P3 (r0 , ϕ) ln |σ − z|dϑ + const (σ = r0 e ).
0
References
[1] I.N. Vekua. Shell Theory: General Methods of Construction. Pitman Advanced
Publishing Program, Boston-London-Melburne, 1985, 287p.
[2] A. Bitsadze. Boundary Value Problems for Second Order Elliptic Equations. Moscow,
Nauka, 1966, 203p (in Russian)
[3] P.G. Ciarlet. Mathematical Elasticity. V.I three-dimensional elasticity. NorthHolland Publishing, Amsterdam-New-Oxford-York-Tokyo, 1998, 471p.
[4] A.L. Goldenveizer. Theory of Elastic thin Shells. Moscow, Nauka, 1976, 512p. (in
Russian)
[5] T.V. Meunargia. A small parameter method for I. Vekua’s nonlinear and non-shallow
shells. Proceedings of the IUTAM Symposium on Relation of Shell, Plate, Beam, and 3D
Models Dedicated to Centenary of Ilia Vekua, 23-27 April, 2007, Tbilisi, Georgia. IUTAM
Bookseries, 9 (2008), 155-166
[6] T.V. Meunargia. On the complex representations for the nonlinear and non-shallow
shells. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math., 23 (2009), 7-11
Received 30.11.2009; revised 23.12.2009; accepted 29.12.2009