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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