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178 ALEKSANDR KORJAKIN ET AL. ∂u ∂γ x ε xx = + x3 =Ω xx + x3χxx ∂x ∂x 1 ⎛ ∂v ⎞ x ∂γ ε = + + =Ω + χ R⎝ ⎜ ∂ϕ ⎠ ⎟ R ∂ϕ 3 ϕ ϕϕ w ϕϕ x3 ϕϕ 1 ∂u ∂v ⎛ 1 ∂γ ∂γ ⎞ γ = + + + = Ω + χ R ∂ϕ ∂x ⎜ ⎝R ∂ϕ ∂x ⎟ ⎠ x ϕ xϕ x3 2 xϕ 2x3 xϕ ∂w γ xx = γ 2 3 x + = Ωxx 3 ∂x 1 ⎛∂w ⎞ γ = γ + − v = 2Ω R ⎜ ⎝∂ϕ ⎟ ⎠ ϕx3 ϕ ϕx3 (13) where R is the reference surface radius of the cylinder, ε xx , ε ϕϕ are the normal strains, and γ xϕ , γ xx , γ 3 ϕx are the shear strains. In addition, Ω 3 xx , Ω ϕϕ , Ω xϕ denote the membrane strains, χ xx , χ ϕϕ , χ xϕ are the bending strains and Ω xx , Ω 3 ϕx are the 3 transverse shear strains in the reference surface, respectively. The general strain-displacement relations for conical shells (see Figure 3) are given by ∂u ∂γ ε = + =Ω + χ ∂x ∂x x xx x3 xx x3 xx 1 ⎛ ∂v ⎞ ε ϕϕ = usin wcos r ⎜ β+ + β ⎝ ∂ϕ ⎟ ⎠ x3 ⎡ ∂γ ϕ cosβ⎛ ∂v ⎞⎤ + ⎢γx sinβ+ + + usinβ+ wcosβ ⎥ =Ω ϕϕ + x3χ r ∂ϕ r ⎜ ⎝∂ϕ ⎟ ⎣ ⎠⎦ ϕϕ 1 ∂u ∂v v γ xϕ = + − sinβ r ∂ϕ ∂x r ⎛1∂γ ∂γ x ϕ γ ϕ cosβ∂v⎞ + x3⎜ + − sinβ+ = 2Ω xϕ + 2x3χ ⎝r ∂ϕ ∂x r r ∂x ⎟ ⎠ xϕ (14) ∂w = 2Ω x γ xx = γ 3 x + xx ∂ 3 1 ⎛∂w ⎞ γ = γ + − v cosβ = 2Ω r ⎜ ⎝∂ϕ ⎟ ⎠ ϕx3 ϕ ϕx3 where r = r(z) is the reference surface radius and β is the semi-vertex angle. The generalstrain-displacement relations for spherical shell (see Figure 4) are givenby
Free Damped Vibrations of Sandwich Shells of Revolution 179 Figure 3. Geometry of the sandwich conical shell. Figure 4. Geometry of the sandwich spherical shell.
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