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1+2(
U
1
E
l
U
2
)
/
(
U
1
G
l
U
2
)
1+2(
U
T
1
G
l
U
1
)
1+2(
U
T
Q
l
=
,
1
E
l
U
1
)
/
(
U
T
2
E
l
U
2
)
/
(
U
T
2
G
l
U
2
)
1+2(
u
1
E
r
u
2
)
/
(
u
1
G
r
u
2
)
1+2(
u
1
E
r
u
1
)
/
(
u
1
G
r
u
1
)
1+2(
u
2
E
r
u
2
)
/
(
u
2
G
r
u
2
)
Q
r
=
,
(1.148)
V
1
F
l
C
l
F
l
V
2
1
F
r
C
r
F
r
˙
υ
˙
υ
2
cos
Ψ
l
=
=cos
Ψ
r
=
=
V
1
||
F
l
C
l
F
l
||
V
2
||
F
l
C
l
F
l
||
υ
1
||
F
r
C
r
F
r
||
˙
υ
2
||
F
r
C
r
F
r
˙
||
(1.149)
V
T
1
diag(
Λ
1
,Λ
2
)
V
2
T
=
˙
υ
υ
2
||
υ
1
||
D
λ
||
υ
2
||
D
λ
1
diag(
λ
1
,λ
2
) ˙
,
=
.
V
1
||
D
Λ
||
V
2
||
D
Λ
||
The following Example
1.9
and the following Box
1.23
illustrate this third multiplicative measure
of deformation.
Example 1.9 (Relative angular shear).
Again, we refer to Example
1.3
, and to Example
1.8
in addition, where the isoparametric mapping
f
= id from an ellipsoid-of-revolution
2
2
A
1
,A
1
,A
2
2
2
r
with respect to the
Cauchy-Green deformation tensor and the absolute angular shear has been analyzed. Here, we
aim at relative angular shear. First, by means of Box
1.23
, we are going to compute cos
Ψ
l
and
cos
Ψ
r
from the two sets of left and right curves, namely from the left Cauchy-Green tensor the
right Cauchy-Green tensor. Second, we derive relative angular shear:
Q
l
=
Q
r
=1.
M
l
=
E
to a sphere
M
e
=
S
End of Example.
Box 1.23 (Relative angular shear).
Left Cauchy-Green matrix:
Right Cauchy-Green matrix:
C
r
=
A
1
cos
2
φ
.
C
l
=
r
2
cos
2
Φ
0
r
2
.
0
1
−E
2
sin
2
φ
(1.150)
E
2
)
2
(1
−E
2
sin
2
φ
)
3
A
1
0
(1
−
0
Left angular shear:
Right angular shear:
U
1
C
l
U
2
u
1
C
r
u
2
u
1
C
r
|| u
2
C
r
cos
Ψ
l
=
,
cos
Ψ
r
=
,
U
1
U
2
C
l
||
C
l
cos
Ψ
l
∼
[1
,
0]C
r
0
=0
,
cos
Ψ
r
∼
[1
,
0]C
l
0
=0
,
1
1
π
2
.
π
2
.
cos
Ψ
l
=0
⇔
Ψ
l
=
±
cos
Ψ
r
=0
⇔
Ψ
r
=
±
(1.151)
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