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Free Damped Vibrations of Sandwich Shells of Revolution 175
components of the metric tensor are given by g ij = g i ⋅ g j . With respect to Equation
(1) from Equation (5) the kinematic equations for the strains in the shell can be deduced
[41]:
2
αβ =Ω αβ + 3χ αβ + 3Φ αβ, 2 α3 = 2Ω α3 + 3κα3
e x x e x
(6)
with
and
2 Ω = ε +ε , 2χ =κ +κ
αβ αβ βα αβ αβ βα
λ λ
αβ bk α λβ bk β λα α3
α α
2 Φ = − − , 2Ω = γ +Ψ
λ
α3
α bα λ
κ =ϕ − γ
λ
αβ αβ αβ αβ αβ α λβ αβ αβ αβ
ε = v −b w, κ = k −b ε , k = γ −b
γ
λ
λ
α α α λ α α α λ
Ψ = w, + b v , ϕ = γ , + b γ
(7)
(8)
Here (...) α denotes the covariant differentiation in the metric characterized by the
fundamental tensor components of the midsurface a αβ = a α ⋅ a β , and b αβ are the curvature
tensor components of the midsurface. All quantities in the Equations (7)
and (8) depend on x α only. Neglecting the strains Φ αβ and κ α3 in Equations (6) the
kinematics equations can be simplified:
e =Ω + x χ , 2e
= 2Ω =γ +Ψ
αβ αβ 3 αβ α3 α3
α α
(9)
It should be noted that the strains Φ αβ and κ α3 can be taken into account in the finite
element analysis of the shell, since the number of unknown functions is the
same—six. However, even for shallow shells, the influence of these strains is significant.
Therefore, for simplicity these terms are hereafter omitted.
For the analysis of shells of various shapes it is more convenient to use orthogonal
coordinates, which conicide with the directions of principal curvatures. In this
case the metric properties of the coordinate system associated with the midsurface
of the shell are characterized by the Lamé’s coefficients A 1 and A 2 . These coefficients
are connected with the components of the fundamental tensor of the
midsurface:
a
αβ
2
⎡a11 0 ⎤ ⎡A
1 0 ⎤
= ⎢
0 a
⎥ = ⎢ ⎥
2
⎣ 22 ⎦ ⎢⎣
0 A2
⎥⎦
(10)
The kinematics, Equations (7) and (8), in orthogonal coordinates of principal cur-