Volume 6, 2002
1
ON TWO-DIMENSIONAL ANALOGUES FOR SHELL-LIKE BODIE
T. Meunargia
I. Vekua Institute of Applied Mathematics
I. Javakhishvili Tbilisi State University
The paper deals with the question of reduction of the three-dimensional
problem of the geometrical and physical nonlinear theory of elasticity to the
two-dimensional one, for shell-like elastic bodies.
∗
I. Let R and R denote the radius-vectors before and after deformation of
∗
the bodies Ω and Ω respectively, moreover
∗
R(x1 , x2 , x3 ) = R(x1 , x2 , x3 ) + U (x1 , x2 , x3 ),
∗
∗
∂j R = R j = ∂j R + ∂j U = Rj + ∂j U
∂
∂j =
∂xj
!
,
where U is the displacement vector, (x1 , x2 , x3 ) are the curvilinear coordinates
in the space.
The equation of equilibrium has the form [1]
q
∗
∗
∗
1 ∂ gσi
q
+
Φ
=0
∗
g ∂xi
∗
∗
∗
where g is the discriminant of the metric tensor of the domain Ω, σ i are
∗
,,contravariant stress vectors“, Φ is an external force.
Under repeating indices summation is meant unless otherwise stated. The
Latin letters take the values 1,2,3, while the Greek letters take the values 1,2.
The equilibrium equation can be written as
√
1 ∂ gσ i
+ Φ = 0, σ i = σ ij (Rj + ∂j U ),
√
g ∂xi
where g is the discriminant of the metric tensor of the domain Ω, σ ij are
contravariant components of the stress tensor,
σi =
v
u∗
ug
t ∗ i
σ ,
g
Φ=
v
u∗
u
tg ∗
g
Φ.
The stress-strain relation has the form
σ ij = (E ijmn + E ijmnpq εpq )εmn ,
2
Bulletin of TICMI
where E ijmn and E ijmnpq are tensors of elasticity of the fourth and sixth rank,
respectively, εij are covariant components of the strain tensor, moreover
E ijmn = λg ij g mn + µ(g im g jn + g in g jm ),
E ijmnpq = η1 g ij g mn g pq + η2 g ij (g mp g nq + g mq g np )+
+η3 g mn (g ip g jq + g iq g jp ) + η4 g pq (g im g jn + g in g jm ),
1
εmn = (Rm ∂n U + Rn ∂m U + ∂m U ∂n U ),
2
λ and µ are Lame’s constants of elasticity, and η1 , η2 , η3 , η4 are modules of
elasticity of the second order for isotropic elastic bodies, g ij = Ri Rj are contravariant components of the metric tensor, Ri and Ri are co and contravariant
base vectors.
II. 1. We consider the coordinate system of lines of curvature, which is
connected normally to the midsurface S of the shell Ω, i.e.
R(x1 , x2 , x3 ) = r(x1 , x2 ) + x3 n(x1 , x2 ),
where r and n are radius-vectors and a normal of the S, x3 is the thickness
coordinate −h ≤ x3 ≤ h, h = const is the semi-thickness.
The dependence between covariant and contravariant base vectors of the
shell Ω and the midsurface S, are expressed as follows
Rα = (1 − kα x3 )rα , Rα =
rα
, R3 = R3 = n, (α = 1, 2)
1 − kα x3
(on α no summation!)
where k1 and k2 are main curvatures of the midsurface S, i.e.
Ri = (1 − ki x3 )ri , Ri =
√
g=
ri
aij
, g ij =
,
1 − ki x3
(1 − ki x3 )(1 − kj x3 )
√
a(1 − k1 x3 )(1 − k2 x3 ), a = a11 a22 − a212 ,



aαβ , i = α, j = β;
a = r r =  0, i = 3, j = β, or i = α, j = 3;

1, i = j = 3,
ij
i j
aαβ = rα rβ , aα3 = a3β = 0, a33 = 0, k3 = 0
(on i, j, α, β no summation!).
For the tensor εij we obtain
!
1
∂j U
∂i U
∂j U
∂i U
+ ri
+
,
εij = (1 − ki x3 )(1 − kj x3 ) rj
2
1 − ki x3
1 − kj x3 1 − ki x3 1 − kj x3
Volume 6, 2002
3
(on i, j no summation!) i.e.
∂β U
∂α U
∂α U
∂β U
1
+ rα
+
)=
εαβ = (1 − kα x3 )(1 − kβ x3 )(rβ
2
1 − kα x3
1 − kβ x3 1 − kα x3 1 − kβ x3
= (1 − kα x3 )(1 − kβ x3 )eαβ ,
1
∂α U
∂α U
εα3 = (1 − kα x3 )(n
+ rα ∂3 U + ∂3 U
)=
2
1 − kα x3
1 − kα x3
= (1 − kα x3 )eα3 ,
1
ε33 = n∂3 U + (∂3 U )2 = e33 .
2
2. Now, we assume the validity of the representations:
∂1 U = (1 − k1 x3 )∂1 V (x1 , x2 ),
∂2 U = (1 − k2 x3 )∂2 V (x1 , x2 ),
∂3 U = Vb (x1 , x2 ),
where V and Vb are the two-dimensional vectors of x1 , x2 .
Taking into consideration the condition
∂1 ∂2 U = ∂2 ∂1 U ⇒ ∂2 (k1 ∂1 V ) = ∂1 (k2 ∂2 V ),
∂3 ∂1 U = ∂1 ∂3 U ⇒ ∂1 Vb = −k1 ∂1 V,
∂3 ∂2 U = ∂2 ∂3 U ⇒ ∂2 Vb = −k2 ∂2 V,
for V (x1 , x2 ) we obtain the following equation
(k1 − k2 )
∂k1 ∂V
∂k2 ∂V
∂ 2V
+
−
= 0.
∂x1 ∂x2 ∂x2 ∂x1 ∂x1 ∂x2
Now, from the system of Gauss equations
√
∂ ln a11
∂k1
= (k2 − k1 )
,
∂x2
∂x2
√
∂ ln a22
∂k2
= (k1 − k2 )
,
∂x1
∂x1
we have
"
#
√
√
∂ ln a11 ∂V
∂ ln a22 ∂V
∂2V
(k1 − k2 )
−
−
= 0.
∂x1 ∂x2
∂x2 ∂x1
∂x1 ∂x2
The general solution of this equation has the form [2]
Zx1
x2
Z
∂R(x01 , t, x1 , x2 )
∂R(t, x02 , x1 , x2 )
dt− v(τ )
dτ.
V (x1 , x2 ) = u(x1 )+v(x2 )− u(t)
∂t
∂τ
0
0
x1
x2
4
Bulletin of TICMI
where R(t, τ, x1 , x2 ) is a Riemann function, u(x1 ) and v(x2 ) are arbitrary vectors.
For the vector U (x1 , x2 , x3 ) we obtain
U (x1 , x2 , x3 ) =
Zx2
∂V (x1 , x2 )
dx1 +
∂x1
Zx2
∂V (x01 , x2 )
dx2 +
∂x2
x01
+
x20
[1 − x3 k1 (x1 , x2 )]
[1 − x3 k2 (x01 , x2 )]
+ (x3 − x03 )Vb (x01 , x02 ) + U (x01 , x02 , x03 ),
and
Vb
=−
Zx1
x01
∂V
dx2 −
k1 (x1 , x2 )
∂x1
Zx2
x02
k2 (x01 , x2 )
∂V (x01 , x2 )
dx2 + Vb (x01 , x02 ).
∂x2
Now for emn we have the following two-dimensional expressions:
1
eαβ = (rα ∂β V + rβ ∂α V + ∂α V ∂β V ),
2
1
eα3 = (n∂α V + rα V Vb + Vb ∂α V ),
2
1
e33 = nVb + Vb 2 .
2
i
The ,,contravariant stress vector“ σ has the form
σ i = (1 − kj x3 )σ ij (rj + ∂j v) =
Ti
1 − ki x3
(on i no summation!), where
T i = (M ijmn + M ijmnpq epq )emn (rj + ∂j V ),
∂j V =
Here
(
∂α V, j = α,
Vb , j = 3.
M ijmn = λaij amn + µ(aim ajn + ain ajm )
M ijmnpq = η1 aij amn apq + η2 aij (apm anq + amq anp )+
+η3 amn (aip ajq + aiq ajp ) + η4 apq (aim ajn + ain ajm ).
At last, we obtain the following two-dimensional equation of equilibrium:
√
!
√
1 ∂ a(1 − k2 x3 )T 1 ∂ a(1 − k1 x3 )T 2
√
+
+
a
∂x1
∂x2
Volume 6, 2002
+
5
∂(1 − k1 x3 )(1 − k2 x3 ) 3
T + (1 − k1 x3 )(1 − k2 x3 )Φ = 0,
∂x3
where
T α = (M αβmn + M αβmnpq epq )emn (rβ + ∂β V )+
+(M α3mn + M α3mnpq epq )emn (n + Vb ),
T 3 = (M 3βmn + M 3βmnpq epq )emn (rβ + ∂β V )+
+(M 33mn + M 33mnpq epq )emn (n + Vb ).
3. Let us consider the boundary condition for the stresses.
∗
The stress vector σ ∗ acting onto area with the mormal (l) has the form
l
σ
∗
∗
∗
∗ ∗
= σ i l i (l i = l R i ).
∗
(l)
∗
The normal l after deformation can be defined as
∗
s
∗
l=
∗
∗
|s
∗
×s
1
1
2
∗
× s 2|
,
∗
∗
where s 1 and s 2 are unit tangent vectors of the boundary surface ∂ Ω ,with
the surface element
∗
∗
∗
∗
∗
d S = | s 1 × s 2 |d s 1 s 2 .
Then we have
1
∗
l=
∗
|s
∗
× s 2|
1


∗
dR
∗
ds
1
∗
×

dR 
= ∗
∗
ds 2
|s
dxi dxj ds1 ds2
=Ri×Rj
=
ds1 ds2 d S∗
∗
=
v
u∗
ug √
t
=
g
v
u∗
ug
t
g
∗
∗
g ∈ijk R k
dxi dxj ds1 ds2
=
ds1 ds2 d S∗
∗
(s1 × s2 )Rk R k
ds1 ds2
∗
dS
∗
⇒l=
∗
∗ ∗
g
1
q
∗
× s 2|

∗
∗
g ∈ijk R k
v
u∗
u
tg
g
∗
∗

d R d R  ds1 ds2
=
×
ds1
ds2 d s∗ 1 d s∗ 2
dxi dxj ds1 ds2
=
ds1 ds2 d S∗
∗
(Ri × Rj )Rk R k
v
u∗
u
tg
dxi dxj ds1 ds2
=
ds1 ds2 d S∗
∗ |s1 × s2 |ds1 ds2
s1 × s2
⇒
Rk R k
∗
g |s1 × s2 |
dS
=
v
u∗
u
tg
1

∗
(lRk ) R k
⇒ li = lRi =
v
u∗
u
tg
li
dS
g d S∗
dS
∗
dS
⇒
(li = lRi ),
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Bulletin of TICMI
s1 × s2
is the normal of the boundary surface before deformation,
|s1 × s2 |
dS is the element of this surface,
where l =
dS = |s1 × s2 |ds1 ds2 ,
∈ijk are the Levi-Civita symbols.
Now the strees vector can be written as
∗ ∗
∗
σ∗ =σli =
(l)
i.e.,
v
u∗
ug ∗
t
σil
g
i
dS
∗
dS
= σ i li
dS
∗
,
dS
∗
dS
σ∗
= σ i li = σ α lα + σ 3 l 3 ,
( l ) dS
∗
where
lα = lRα , l3 = ln.
On the surfaces x3 = ±h we have l = n and so
σ (n) (x1 , x2 , ±h) = σ 3 (x1 , x2 , ±h).
The stress vector σ
normal lb has the form
∗
(l)
acting on the lateral surface dSb = dsbdx3 with the
b ).
σ (bl) = σ α (lR
α
The normal lb before deformation can be defined as:
where
lb =
dR
× n,
dsb
dR ds
dn
dR
d(r + x3 n) ds
= sb =
=
= s + x3
dsb
ds dsb
ds
dsb
ds
Therefore,
⇒ sb = [(1 − k3 x3 )s + τs x3 l]
ds
.
dsb
ds
,
dsb
(lb × sb = n, l × s = n).
lb = [(1 − k3 ks )l − x3 τs s]
!
ds
⇒
dsb
b sb and l, s are the unit vectors of the tangential normal and tangent
where l,
of the lateral curve of the surfaces x3 = const and x3 = 0 (midsurface),
respectively, ks and τs are the normal curvature and geodesic torsion of the
midsurface, dsb and ds are linear elements of the surfaces x3 = const and x3 = 0,
respectively, moreover
dsb =
q
1 − 2x3 ks + x32 (ks2 + τs2 )ds ⇒
Volume 6, 2002
⇒ dsb =
v
u
u
t
a11 (1 − k1 x3
)2
dx1
ds
!2
+ a22 (1 − k2 x3
7
)2
dx2
ds
!2
ds.
On the other hand, we have [3]
lb =
dxα ds √
dR
dR
ds
dxα ds
×n=
× n = Rα × n
= g ∈α3β Rβ
=
dsb
ds
dsb
ds dsb
ds dsb
=
Therefore,
r
r
g√
g
dxα ds
dxα ds
a ∈α3β Rβ
(rα × n)rβ Rβ
=
=
a
ds dsb
a
ds dsb
r
r
g
g
ds
β ds
=
⇒
=
(s × n)rβ R
(lrβ )Rβ
a
dsb
a
dsb
r
g ds
b
b
⇒ lRβ = lβ =
lβ .
a dsb
σ (bl) =
σ α lbα
=
r
g α ds
σ lα ⇒
a
dsb
ds
= (1 − k1 x3 )(1 − k2 x3 )(σ 1 l1 + σ 2 l2 ) ⇒
dsb
ds
σ (bl) = (1 − k2 x3 )T 1 l1 + (1 − k1 x3 )T 2 l2 .
dsb
Thus, we obtain the following system of two-dimensional equations of the
geometrically and physically non-linear theory for shell-like elastic bodies:
a) Equilibrium equations
√
!
√
1 ∂ a(1 − k2 x3 )T 1 ∂ a(1 − k1 x3 )T 1
√
+
− 2(H − Kx3 )T 3 + F = 0
a
∂x1
∂x2
⇒ σ (bl)
(0)
2H = k1 + k2 , K = k1 k2 , (1 − k1 x3 )(1 − k2 x3 )Φ = F = F +x3 F
b) Stress-strain relation
T i (x1 , x2 ) = (M iβmn + M iβmnpq epq )emn (rβ + ∂β V )+
where
III. Special cases
+(M i3mn + M i3mnpq epq )emn (n + Vb ),
1
eαβ = (rα ∂β V + rβ ∂α V + ∂α V ∂β V ),
2
1
eα3 = (n∂α V + nVb + Vb ∂α V ),
2
1
e33 = nVb + Vb 2 .
2
!
(1)
;
8
Bulletin of TICMI
1
1. Spherical shell (k1 = k2 = − )
R
The vector of displacement U for the spherical shell has the form
’
U (x1 , x2 , x3 ) = 1 +




’
“
x3
V (x1 , x2 ) ⇒
R
“
x3
∂α U = 1 +
∂α V (α = 1, 2),
R
⇒
 ∂ U = 1 V.

3
R
The equation of equilibrium can be written as:
√
1 ∂ aT α
2
√
+ T 3 + F = 0,
a ∂xα
R
where
T i (x1 , x2 ) = (M ijmn + M ijmnpq epq )emn (rj + ∂j V ) (∂3 V =
1
V ),
R
1
emn (x1 , x2 ) = (rm ∂n V + rn ∂m V + ∂m V ∂n V ),
2
“
’
x3
Φ.
F (x1 , x2 ) = 1 +
R
The stress vector has the form
σ (l)
’
“
x3
ds
= 1+
T α lα = T α lα = T(l) ,
R
dsb
’
’
“
“
x3
ds .
dsb = 1 +
R
2. Cylindrical shell (k1 = − R1 , k2 = 0)
The vector of displacement for the cylindrical shell has the form
’
“
’
“
x3
U (x1 , x2 , x3 ) = 1 +
u(x1 ) + v(x2 ).
R
Then









x3 du(x1 )
∂1 v = 1 +
R
dx1
dv(x2 )
 ∂2 v = dx


2


1


 ∂3 v = u(x1 )
R
The equilibrium equation looks like:
√
1 ∂ aT 1
1
√
+ T 3 = 0,
a ∂x1
R
Volume 6, 2002
9
√
1 ∂ aT 2
√
+ Φ = 0 (Φ = Φ(x1 , x2 )),
a ∂x2
where
i
T = (M
+(M
i1mn
i2mn
+M
+M
i1mnpq
i2mnpq
du(x1 )
r1 +
dx1
epq )emn
epq )emn
!
!
’
1
u(x1 ) .
R
dv(x2 )
r2 +
dx2
+(M i3mn + M i3mnpq epq )emn n +
“
Here
M ijmn = λδ ij δ mn + µ(δ im δ jn + δ in δ jm ),
M ijmnpq = η1 δ ij δ mn δ pq + η2 δ ij (δ mn δ nq + δ mn δ np )+
+η3 δ mn (δ ip δ jq + δ iq δ jp ) + η4 δ pq (δ im δ jn + δ in δ jm ),
ij
δ =
For the eij we obtain
e12
e11
(
1, i = j
0, i =
6 j
!
.
du(x1 )
du dv
dv(x2 )
1
r1
=
+ r2
+
2
dx1
dx2
dx1 dx2
du
1
= r1
+
dx1 2
!
!
!
e13
r1 u
u du
1
du
n
+
+
=
,
2
dx1
R
R dx1
e23
dv
r2 u
1
u dv
=
n
+
+
,
2
dx2
R
R dx2
du
dx1
!2
, e22
dv
1
+
= r2
dx2 2
dv
dx2
!2
= e21 ,
, e33 =
nu
u2
.
+
R
2R2
The stress vector σ (bl) has the form (lb = r2 ):
σ (bl) = σ (r2 ) = (1 +
x3 2 ds
)T l2 = T 2 .
R
dsb
REFERENCES
[1] Vekua I.N., Shell Theory: General Methods of Construction. Moscow, Nauka, 1982
(Russian)
[2] Vekua I.N., New Methods of Solution of Elliptic Equations. Moscow, Gostekhizdat,
1948 (Russian)
[3] Meunargia T.V., On The Method of Geometrically Nonlinear Theory of Non-Shallow
Shells. Proceedings of A.Razmadze Mathematical Institute, vol.119, 1998
Received June 21, 2002; revised October 20, 2002; accepted December 5, 2002.