Appendix A: Geometric Quantities
165
b αβ =
⎡
⎢
⎣
−
1
R
0
0 −R sin
2
Θ
1
R
⎤
⎥
⎦ , b
β
α =
⎡
⎢
⎣
−
1
R
0
0 −
1
R
⎤
⎥
⎦ .
(A.37)
The components of the shifter tensor are
μ
β
α = δ
β
α − Θ
3
· b
β
α =
⎡
⎢
⎣
1 +
Θ
3
R
0
0
1+
Θ
3
R
⎤
⎥
⎦ .
(A.38)
The Christoffel symbols of the second kind for the point at the mid-surface are
Γ
1
αβ =
0
0
0 −R sin
Θ
1
R
cos
Θ
1
R
, Γ
2
αβ =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0
cos
Θ
1
R
R sin
Θ 1
R
cos
Θ
1
R
R sin
Θ 1
R
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(A.39)
Therefore, the covariant derivatives and the abbreviations
n
ϕ αβ can be obtained as
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
n
v 1|1 =
n
v 1,1
n
v 1|2 =
n
v 1,2 −
cos
Θ 1
R
R sin
Θ 1
R
n
v 2
n
v 2|1 =
n
v 2,1 −
cos
Θ 1
R
R sin
Θ 1
R
n
v 2
n
v 2|2 =
n
v 2,2 + R sin
Θ 1
R
cos
Θ 1
R
n
v 1
and
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
n
ϕ 11 =
n
v 1,1 +
1
R
n
v 3
n
ϕ 12 =
n
v 1,2 −
cos
Θ 1
R
R sin
Θ 1
R
n
v 2
n
ϕ 21 =
n
v 2,1 −
cos
Θ 1
R
R sin
Θ 1
R
n
v 2
n
ϕ 22 =
n
v 2,2 + R sin
Θ 1
R
cos
Θ 1
R
n
v 1
+ R sin
2
Θ 1
R
n
v 3
n
ϕ 31 =
n
v 3,1 −
1
R
n
v 1
n
ϕ 32 =
n
v 3,2 −
1
R
n
v 2
(A.40)
We introduce several variables that are frequently used in the strain-displacement
expressions, as shown in Table A.1.