Geography Reference
In-Depth Information
1-1 Cauchy-Green Deformation Tensor
A first multiplicative measure of deformation: the Cauchy-Green deformation tensor, polar
decomposition, singular value decomposition, Hammer retroazimuthal projection.
There are various local multiplicative and additive measures of deformation being derived from
the infinitesimal distances d
S
2
of
2
l
and d
s
2
of
r
, with
M
M
d
S
2
=
G
MN
(
U
L
)d
U
M
d
U
N
versus d
s
2
=
g
μν
(
u
λ
)d
u
μ
d
u
ν
.
(1.12)
The mapping of type deformation,
f
:
M
l
→
M
r
, is
r
epresented locally by
f
, in particular
u
μ
, the mapping of type inverse deformation,
f
−
1
:
U
M
2
r
2
→
M
→
M
l
, is represented locally
by
f
−
1
, in particular
u
μ
U
M
, with
U
M
u
μ
=
f
μ
(
U
M
)and
u
μ
U
M
=
F
M
(
u
μ
). In the
→
→
→
2
l
2
2
r
2
left and right tangent bundles
T
M
×
M
l
and
M
×
M
r
, we represent locally the projections
2
l
2
2
2
r
2
2
π
(
T
M
×
M
l
)=
T
M
l
and
π
(
T
M
×
M
r
)=
T
M
r
by the
pullback map
and the
pushforward map
,
in particular, by
f
∗
:d
U
M
=
∂U
M
∂u
μ
∂u
μ
d
u
μ
versus
f
∗
:d
u
μ
=
∂U
M
d
U
M
.
(1.13)
∂U
M
/∂u
μ
∂u
μ
/∂U
M
|
|
>
0versus
|
|
>
0
preserve the orientation ∂/∂U
∧
∂/∂V
and
∂/∂u
∧
∂/∂v
,
2
2
respectively, of
r
, respectively.
The first multiplicative measure of deformation has been introduced by
Cauchy
(
1828
)and
Green
(
1839
) reviewed in the sets of relations shown in Box
1.1
, where the abbreviation Left CG
indicates the
left Cauchy-Green deformation tensor
and the abbreviation Right CG indicates the
right Cauchy-Green deformation tensor
. With respect to the deformation gradients, the left and
right Cauchy-Green tensors are represented in matrix algebra by
M
l
and
M
C
l
:= J
l
G
r
J
l
versus C
r
:= J
r
G
l
J
r
.
(1.14)
The set of
deformation gradients
is described by the two
Jacobi matrices
J
l
and J
r
,whichobey
the matrix relations
J
l
:=
∂u
μ
∂U
M
=J
−
1
J
r
:=
∂U
M
∂u
μ
=J
−
l
.
versus
(1.15)
r
The abstract notation hopefully becomes more concrete when you work yourself through Exam-
ple
1.3
where we compute the Cauchy-Green deformation tensor for an isoparametric mapping
of a point on an ellipsoid-of-revolution to a point on a sphere.
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