158
Appendix A: Geometric Quantities
Fig. A.1 Curvilinear
coordinates for a plate
structure
Θ
3
Θ
1
Θ
2
X
2
X
3
X
1
Using the formulation g
i
= g i j g j one obtains the contravariant base vectors in the
shell space
g
1
=
⎧
⎨
⎩
1
0
0
⎫
⎬
⎭
, g
2
=
⎧
⎨
⎩
0
1
0
⎫
⎬
⎭
, g
3
=
⎧
⎨
⎩
0
0
1
⎫
⎬
⎭
.
(A.5)
The covariant base vectors of the point at the mid-surface are
a 1 =
⎧
⎨
⎩
1
0
0
⎫
⎬
⎭
, a 2 =
⎧
⎨
⎩
0
1
0
⎫
⎬
⎭
, a 3 = n =
a 1 × a 2
1 × a 2
=
⎧
⎨
⎩
0
0
1
⎫
⎬
⎭
.
(A.6)
The covariant and contravariant metric tensors at the mid-surface will be
a i j = a i · a j =
⎡
⎣
1 0 0
0 1 0
0 0 1
⎤
⎦ , a
i j
= [a i j ]
−1
=
⎡
⎣
1 0 0
0 1 0
0 0 1
⎤
⎦ .
(A.7)
The contravariant base vectors at the mid-surface are
a
1
=
⎧
⎨
⎩
1
0
0
⎫
⎬
⎭
, a
2
=
⎧
⎨
⎩
0
1
0
⎫
⎬
⎭
, a
3
= a 3 =
⎧
⎨
⎩
0
0
1
⎫
⎬
⎭
.
(A.8)
The partial derivatives of the covariant base vectors at the mid-surface are
a i, j =
⎧
⎨
⎩
0
0
0
⎫
⎬
⎭
.
(A.9)
The covariant and mixed components of the curvature tensor are
b αβ = a α,β · a 3 =
0 0
0 0
, b
β
α = a
βγ
· b αγ =
0 0
0 0
.
(A.10)
The components of the shifter tensor are