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174 ALEKSANDR KORJAKIN ET AL. rotations at the midsurface = A −a 3 3 (2) In Equation (1) the vectors v and have three components, respectively α α 3, α 3 v= v a + wa = γ a + wa α (3) In this sense, the FOSDT is characterized by 6 independent degrees of freedom (FOSDT-6). The FOSDT version with five degrees of freedom (FOSDT-5) is usually used. The reasons for such limitation is the deformability of the thin-walled structure even in the case of moderate thickness: the magnitude of the assumed drilling of the normal is much smaller in comparison with the other two rotations. So in the case of the FOSDT-5 we assume γ = 0. For shells of revolution in this case the displacements u, v, and w in the x (meridian), ϕ (circumferential) and x 3 (normal, outward from the reference surface of the shell) directions at any point are expressed through the following approximations [31]: ux ( , ϕ , x, t) = u( x, ϕ , t) + xγ ( x, ϕ, t) 3 0 3 vx ( , ϕ , x, t) = v( x, ϕ , t) + xγ ( x, ϕ, t) 3 0 3 wx (, ϕ , x,) t = w(, xϕ,) t 3 0 x ϕ (4) where u 0 , v 0 , w 0 are the displacement components at the reference surface (midsurface) and γ x , γ ϕ are the rotations of the normal associated with the transverse shear deformations. Below the index “0” will be omitted. The approximations, Equation (4), are similar to Timoshenko’s proposal in the beam theory [45] and the analogue in the plate theory [18,30]. STRAIN-DISPLACEMENT RELATIONS The displacements of an arbitrary point of the shell continuum is approximated by two vectors. Therefore, the displacement field of the shell in the FOSDT-6 including extension of the normal is represented by six unknown functions—three displacements (v α ,w) and three rotations (γ α ,γ). These quantities are functions of the shell middle surface coordinates x α only. From these displacements and rotations the two-dimensional strains can be obtained. The three-dimensional Green’s strain tensor in the linear case is given by Reference [16,41]. 2e ij = u, i ⋅ g j + u, j ⋅g i (5) Here (...), i denotes differentiation with respect to the coordinates x i , the dot means scalar product and g i is a triad of vectors for coordinates x i on the surfaces (x 3 = const) parallel to the middle surface of the shell. For these coordinates the
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-
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