Civil Engineering Reference
In-Depth Information
Hence, the metric tensor on Ω can be represented by a symmetrical positive-definite
matrix with coefficients that are functions of the reduced principal curvatures and the pro-
jections of
u
and
v
onto the principal curvature directions
ˆ
ˆ
and
.
Metric tensor
M
on tangent plane T
p
can be extended to 3D space by a simple modification:
v
v
1
2
2
κ
00
0 0
00
v
v
n
ˆ
ˆ
1
1
wherecisanarbitrary constant
M
= ˆˆ
vvn
2
κ
3
12
2
2
c
In terms of M
3D
, the new M
Ω
is given by
2
κ
0
00
v
v
n
ˆ
ˆ
1
1
u
v
u
v
MM
D
=
=
ˆˆ
vvn
uv
2
uv
κ
0
3
3
D
12
2
2
00
c
κ
2
00
uv vv
uv vv
un vn
⋅
ˆ
⋅
ˆ
1
1
1
uv uv un
vv vv vn
⋅
ˆ
⋅
ˆ
⋅
1
2
M
=
κ
2
⋅
ˆ
⋅
ˆ
0
0
00c
3
⋅
ˆ
⋅
ˆ
⋅
2
2
2
1
2
⋅
⋅
2
2
2
2
2
κ
(
uv
⋅ +⋅ +⋅
ˆ )
κ
(
uv
ˆ )
c
(
un
)
m
1
1
2
2
12
=
2
2
2
2
2
m
κ
(
vv
⋅ +⋅ +⋅
ˆ )
κ
(
vv
ˆ )
c
(
vn
)
21
1
1
2
2
with
mm
)( )
.
For vector V on T
p
, V ⋅
n
= 0, and there is no difference for vectors on T
p
measured by either
M
Ω
or M
Ω3D
. However, if A and B are two distinct points on a planar domain Ω, and ϕ(A) and
ϕ(B) are two points on surface S such that edge V = ϕ(A)ϕ(B) is not in S and does not lie on
the tangent planes T
ϕ(A)
or T
ϕ(B)
, then V ⋅
n
≠ 0 and in fact V ⋅
n
accounts for the contribution
of the component of V out of the tangent plane T
p
towards the length measure. The param-
eter c is used to adjust the significance of the out-of-plane component in length measure.
==⋅
κ
2
(
uv vv
ˆ )(
⋅
ˆ )
+
κ
2
(
uv vv
⋅
ˆ )(
⋅ +
ˆ )(
c
u
⋅
nvn
⋅
21
12
1
1
1
2
2
2
4.2.7 Metric tensor and Green-Cauchy deformation tensor
The metric tensor M can be interpreted in terms of the Green-Cauchy deformation tensor C
(Marsden and Hughes 1983) in which the length of infinitesimal fibre d
x
upon deformation
is given by ds
2
= d
x
⋅ C ⋅ d
x
, in which ds is the deformed length of d
x
. Based on the length
measure by metric tensor M for vector d
x
, we have dr
2
= d
x
⋅ M ⋅ d
x
.
From this, we can see that M is identical to C mathematically, although they have different
physical interpretations; dr represents the length of d
x
as measured with respect to metric M,
and ds represents the length of d
x
after deformation. Many results established in the theory
of large deformation can be applied to the geometrical quantities governed by the metric M.
Metric tensor M can be expressed in terms of the principal base vectors as follows.
2
ˆ
ˆ
λ
0
e
e
1
1
T
T
T
M
=
ˆˆ
=
R UU
RRURURFF
=
(
)
=
ee
12
2
0
λ
2
2
