3.2 Mathematical Preliminaries
39
X
2
r
¯ r
¯
R
0
u
R
g 2
u
Θ
2
Θ
1
¯ g
1
¯ g
2
n
X
3
Θ
3
a 1
X
1
Θ
1
g 3
g 1
a 2
Θ
2
mid-surface
¯ a 2
¯ a 1
n
¯ a 3
Θ
3
¯ g
3
¯ a 3
¯
P V
P Ω
P V
¯
P Ω
1
u
a 2
a 1
¯ n
Fig. 3.1 Definition of base vectors
coordinate system is usually fixed, while the convective coordinate system is set on
structures. The convective coordinate system can be plate, cylindrical, spherical or
any other coordinates. The position vector of an arbitrary point (P V ) in the shell space
is denoted by R(Θ
1
, Θ
2
, Θ
3
), while r(Θ
1
, Θ
2
) refers to that of an arbitrary point
(P ) at the mid-surface.
In order to present the structural deformation, two configurations are defined,
namely the undeformed configuration and the deformed configuration, as shown in
Fig. 3.1. The undeformed configuration is shown in the left part of the figure, while
the deformed configuration is shown in the middle part of the figure. Furthermore,
the right hand side of the figure shows the rotation of the Θ
3 -line. An arbitrary point
in the shell space and at the mid-surface is denoted by P V and P , respectively. In
this report, the Latin indices vary from 1 to 3, whereas the Greek indices only take
1 or 2.
3.2.2 Base Vectors and Metric Tensor in Shell Space
Considering an arbitrary point P V in the undeformed shell space, the covariant base
vectors g i are defined as the tangent of the coordinate lines, expressed by
g i =
∂ R
∂Θ i = R ,i ,
(3.1)
where the subscript “, i” represents the spatial derivative with respect to Θ
i . Because
the coordinate lines can be arbitrarily defined, the base vectors g i may not be perpendicular with each other, like the Cartesian coordinate system. To avoid complex
computation problems, we introduce contravariant base vectors g
i , which are determined by means of the vector products of covariant base vectors
39
X
2
r
¯ r
¯
R
0
u
R
g 2
u
Θ
2
Θ
1
¯ g
1
¯ g
2
n
X
3
Θ
3
a 1
X
1
Θ
1
g 3
g 1
a 2
Θ
2
mid-surface
¯ a 2
¯ a 1
n
¯ a 3
Θ
3
¯ g
3
¯ a 3
¯
P V
P Ω
P V
¯
P Ω
1
u
a 2
a 1
¯ n
Fig. 3.1 Definition of base vectors
coordinate system is usually fixed, while the convective coordinate system is set on
structures. The convective coordinate system can be plate, cylindrical, spherical or
any other coordinates. The position vector of an arbitrary point (P V ) in the shell space
is denoted by R(Θ
1
, Θ
2
, Θ
3
), while r(Θ
1
, Θ
2
) refers to that of an arbitrary point
(P ) at the mid-surface.
In order to present the structural deformation, two configurations are defined,
namely the undeformed configuration and the deformed configuration, as shown in
Fig. 3.1. The undeformed configuration is shown in the left part of the figure, while
the deformed configuration is shown in the middle part of the figure. Furthermore,
the right hand side of the figure shows the rotation of the Θ
3 -line. An arbitrary point
in the shell space and at the mid-surface is denoted by P V and P , respectively. In
this report, the Latin indices vary from 1 to 3, whereas the Greek indices only take
1 or 2.
3.2.2 Base Vectors and Metric Tensor in Shell Space
Considering an arbitrary point P V in the undeformed shell space, the covariant base
vectors g i are defined as the tangent of the coordinate lines, expressed by
g i =
∂ R
∂Θ i = R ,i ,
(3.1)
where the subscript “, i” represents the spatial derivative with respect to Θ
i . Because
the coordinate lines can be arbitrarily defined, the base vectors g i may not be perpendicular with each other, like the Cartesian coordinate system. To avoid complex
computation problems, we introduce contravariant base vectors g
i , which are determined by means of the vector products of covariant base vectors