Section 2.10
Solid Mechanics Part III
Kelly
279
2.10 Convected Coordinates
In this section, the deformation and strain tensors described in §2.2-3 are now described
using convected coordinates (see §1.16).
Note that all the tensor relations expressed in
symbolic notation already discussed, such as
C
U
=
,
i
i
i
n
N
F
λ
=
ˆ
,
lF
F
=
&
, are independent
of coordinate system, and hold also for convected coordinates.
2.10.1
Convected Coordinates
Introduce the curvilinear coordinates
i
Θ
.
The material coordinates can then be written as
)
,
,
(
3
2
1
Θ
Θ
Θ
=
X
X
(2.10.1)
so
i
i
X
E
X
=
and
i
i
i
i
d
dX
d
G
E
X
Θ
=
=
,
(2.10.2)
where
i
G
are the covariant base vectors in the reference configuration, with corresponding
contravariant base vectors
i
G
, Fig. 2.10.1, with
i
j
j
i
δ
=
⋅
G
G
(2.10.3)
Figure 2.10.1: Curvilinear Coordinates
The coordinate curves, curves of constant
i
Θ
, form a net in the undeformed configuration.
One says that the curvilinear coordinates are
convected
or
embedded
, that is, the coordinate
curves are attached to material particles and deform with the body, so that each material
1
1
,
x
X
2
2
,
x
X
3
3
,
x
X
X
1
1
,
e
E
2
2
,
e
E
1
g
2
g
current
configuration
reference
configuration
1
G
2
G
x

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