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184 ALEKSANDR KORJAKIN ET AL.
pressed as follows:
(1) (1) (1) (1)
1 3 x , 1 3 ϕ , 1
u = u + x γ v = v + x γ w = w
(26)
For the third (top) layer the displacements are represented as
(3) (3) (3) (3)
3 3 x , 3 3 ϕ , 3
u = u + x γ v = v + x γ w = w
(27)
The displacement continuity conditions between the layers of the sandwich shell
are given as
u = u , v = v , w = w ( i = 1,2)
i i+ 1 i i+ 1 i i+
1
(28)
Taking into account these displacement continuity conditions, the displacements
for the core can be presented as
(2) (2) (2) (2)
2 3 x , 2 3 ϕ , 2
u = u + x γ v = v + x γ w = w
(29)
And the variables u (2) , v (2) , γ
( 1) ( 1) ( 3) ( )
γ , γ , γ , γ
3 as follows:
x
ϕ
x
ϕ
( 2 ) ( )
x γ 2
ϕ
and depend from u (1) , v (1) , u (3) , v (3) ,
(2) (1) (3) (1) (3)
=+ 1 + 2 + 3γx
− 3γx
u Fu F u F F
(2) (1) (3) (1) (3)
x Fu 4 Fu 4 F2 x F1
x
γ = − + + γ + γ
(2) (1) (3) (1) (3)
=+ 1 + 2 + 3γϕ
− 3γϕ
v Fv F v F F
(2) (1) (3) (1) (3)
ϕ Fv 4 Fv 4 F2 ϕ F1
ϕ
γ = + − − γ − γ
(30)
where
(1) (1)
h
h
1
1 = 1 + 2, 2 = − ,
(2) 3 = 1 , 4 =
(2)
F F F F F F
2h
2
The displacement field of the sandwich shell can be described by vector components
h
u
T
(1) (1) (1) (1) (1) (3) (3) (3) (3)
0 0 0 x ϕ 0 0 x ϕ
= { u , v , w , γ , γ , u , v , γ , γ }
(31)
For the case of a six nodal point finite element (with 9 degrees of freedom per
node), the displacement variation over the finite element can be described by