1-1 Cauchy–Green deformation tensor
5
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 dS
2 of M
2
l and ds
2 of M
2
r , with
dS
2 = G MN (U
L )dU
M dU
N
versus ds
2 = g µν (u
λ )du
µ du
ν .
(1.12)
The mapping of type deformation, f : M
2
l → M
2
r , is represented locally by f , in particular U
M
→ u
µ ,
the mapping of type inverse deformation, f
−1 : M
2
r → M
2
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 left and right tangent
bundles T M
2
l × M
2
l and T M
2
r × M
2
r , we represent locally the projections π(T M
2
l × M
2
l ) = T M
2
l and
π(T M
2
r × M
2
r ) = T M
2
r by the pullback map and the pushforward map, in particular, by
f ∗ : dU
M =
∂U
M
∂u µ du
µ versus f
∗ : du
µ =
∂u
µ
∂U M dU
M .
(1.13)
∂U
M /∂u
µ
> 0 versus
∂u
µ /∂U
M
> 0 preserve the orientation ∂/∂U ∧ ∂/∂V and ∂/∂u ∧ ∂/∂v,
respectively, of M
2
l and M
2
r , respectively.
The first multiplicative measure of deformation has been introduced by A. L. Cauchy (1828) and
G. 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
C l := J
T
l G r J l versus C r := J
T
r G l J r .
(1.14)
The set of deformation gradients is described by the two Jacobi matrices J l and J r , which obey the
matrix relations
J l :=
∂u
µ
∂U
M
= J
−1
r
versus J r :=
∂U
M
∂u
µ
= J
−1
l
.
(1.15)
The abstract notation hopefully becomes more concrete when you work yourself through Example 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.
Box 1.1 (Left and right Cauchy–Green deformation tensor).
Left CG:
Right CG:
ds
2 =
d S
2 =
= g µν
˘
f
λ (U
L )
¯ ∂u
µ
∂U M
∂u
ν
∂U N dU
M dU
N =
= G MN
˘
F
L (u
λ )
¯ ∂U
M
∂u
µ
∂U
N
∂u
ν du
µ du
ν =
= c MN (U
L ) dU
M dU
N ,
= C µν (u
λ )du
µ du
ν ,
c MN (U
L ) =
C µν (u
λ ) =
= g µν (U
L )
∂u
µ
∂U
M
(U
L )
∂u
ν
∂U
N
(U
L ) .
= G MN (u
λ )
∂U
M
∂u
µ (u
λ )
∂U
N
∂u
ν (u
λ ) .
(1.16)
5
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 dS
2 of M
2
l and ds
2 of M
2
r , with
dS
2 = G MN (U
L )dU
M dU
N
versus ds
2 = g µν (u
λ )du
µ du
ν .
(1.12)
The mapping of type deformation, f : M
2
l → M
2
r , is represented locally by f , in particular U
M
→ u
µ ,
the mapping of type inverse deformation, f
−1 : M
2
r → M
2
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 left and right tangent
bundles T M
2
l × M
2
l and T M
2
r × M
2
r , we represent locally the projections π(T M
2
l × M
2
l ) = T M
2
l and
π(T M
2
r × M
2
r ) = T M
2
r by the pullback map and the pushforward map, in particular, by
f ∗ : dU
M =
∂U
M
∂u µ du
µ versus f
∗ : du
µ =
∂u
µ
∂U M dU
M .
(1.13)
∂U
M /∂u
µ
> 0 versus
∂u
µ /∂U
M
> 0 preserve the orientation ∂/∂U ∧ ∂/∂V and ∂/∂u ∧ ∂/∂v,
respectively, of M
2
l and M
2
r , respectively.
The first multiplicative measure of deformation has been introduced by A. L. Cauchy (1828) and
G. 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
C l := J
T
l G r J l versus C r := J
T
r G l J r .
(1.14)
The set of deformation gradients is described by the two Jacobi matrices J l and J r , which obey the
matrix relations
J l :=
∂u
µ
∂U
M
= J
−1
r
versus J r :=
∂U
M
∂u
µ
= J
−1
l
.
(1.15)
The abstract notation hopefully becomes more concrete when you work yourself through Example 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.
Box 1.1 (Left and right Cauchy–Green deformation tensor).
Left CG:
Right CG:
ds
2 =
d S
2 =
= g µν
˘
f
λ (U
L )
¯ ∂u
µ
∂U M
∂u
ν
∂U N dU
M dU
N =
= G MN
˘
F
L (u
λ )
¯ ∂U
M
∂u
µ
∂U
N
∂u
ν du
µ du
ν =
= c MN (U
L ) dU
M dU
N ,
= C µν (u
λ )du
µ du
ν ,
c MN (U
L ) =
C µν (u
λ ) =
= g µν (U
L )
∂u
µ
∂U
M
(U
L )
∂u
ν
∂U
N
(U
L ) .
= G MN (u
λ )
∂U
M
∂u
µ (u
λ )
∂U
N
∂u
ν (u
λ ) .
(1.16)