Appendix A: Geometric Quantities
159
μ
β
α = δ
β
α − Θ
3 b
β
α =
1 0
0 1
.
(A.11)
The Christoffel symbols of the second kind for the point at the mid-surface are
Γ
1
αβ = Γ
2
αβ =
0 0
0 0
.
(A.12)
Therefore, the covariant derivatives and the abbreviations
n
ϕ αβ can be obtained as
n
v α|β =
n
v α,β ,
n
ϕ αβ =
n
v α,β ,
n
ϕ 3α =
n
v 3,α .
(A.13)
A.2 Cylindrical Structure
The Cartesian coordinate system (X
1 , X
2 , X
3 ) and the curvilinear coordinate system
(Θ
1 , Θ
2 , Θ
3 ) of a cylindrical structure are shown in Fig. A.2.
The curvilinear coordinates are defined as
Θ
1
= −z, Θ
2
= α, Θ
3
= r − R.
(A.14)
Here, R denotes the radius of the mid-surface, and r represents the radius of an
arbitrary large cylindrical surface. The position vectors of an arbitrary point in the
shell space and at the mid-surface are respectively expressed as
R =
⎧
⎨
⎩
(R + Θ
3
) cos
Θ
2
(R + Θ
3
) sin
Θ
2
−Θ
1
⎫
⎬
⎭
, r =
⎧
⎨
⎩
R cos
Θ
2
R cos
Θ
2
−Θ
1
⎫
⎬
⎭
.
(A.15)
Fig. A.2 Curvilinear
coordinates for a cylindrical
structure
X
1
Θ
1
Θ
2
α
Θ
3
X
2
X
3