1-8 Relative angular shear 41
In Box 1.22, we have collected various representations of angular shear, in particular, in terms of the
Cauchy–Green deformation and Euler–Lagrange deformation tensors, as well as their eigenvalues.
Box 1.22 (Left and right angular shear).
cos Ψ l =
˙
U
T
1 G l ˙
U 2
˙
U 1 G l ˙
U 2 G l
=
c o sΨ r =
˙
u
T
1 G r ˙
u 2
˙
u 1 Gr ˙
u 2 Gr
=
=
˙
u
T
1 C r ˙
u 2
˙
u 1 C r ˙
u 2 C r
,
=
˙
U
T
1 C l ˙
U 2
˙
U 1 C l ˙
U 2 C l
,
(1.145)
Q l :=
cos Ψ r
cos Ψ l
=
˙
U
T
1 C l ˙
U 2
˙
U
T
1 G l
˙
U 2
×
Q r :=
cos Ψ l
cos Ψ r
=
˙
u
T
1 C r ˙
u 2
˙
u
T
1 G r ˙
u 2
×
×
˙
U 1 G l ˙
U 2 G l
˙
U 1 C l ˙
U 2 C l
=
×
˙
u 1 Gr ˙
u 2 Gr
˙
u 1 Cr ˙
u 2 Cr
=
=
˙
U
T
1 C l ˙
U 2
˙
U
T
1 G l
˙
U 2
1
Λ( ˙
U 1 )Λ( ˙
U 2 )
,
=
˙
u
T
1 C r ˙
u 2
˙
u
T
1 G r ˙
u 2
1
λ( ˙
u 1 )λ( ˙
u 2 )
,
(1.146)
Q l =
˙
U
T
1
`
2E l + G l
´ ˙
U 2
˙
U
T
1 G l
˙
U 2
˙
U 1 G l
q
˙
U
T
1
`
2E l + G l
´ ˙
U 1
×
Q r =
˙
u
T
1
`
2E r + G r
´
˙
u 2
˙
u
T
1 G r ˙
u 2
˙
u 1 Gr
q
˙
u
T
1
`
2E r + G r
´
˙
u 1
×
×
˙
U 2 G l
q
˙
U
T
2
`
2E l + G l
´ ˙
U 2
,
×
˙
u 2 Gr
q
˙
u
T
2
`
2E r + G r
´
˙
u 2
,
(1.147)
Q l =
1 + 2
` ˙
U
T
1 E l ˙
U 2
´
/
` ˙
U
T
1 G l ˙
U 2
´
q
1 + 2
` ˙
U
T
1 E l
˙
U 1
´
/
` ˙
U
T
1 G l
˙
U 1
´ q
1 + 2
` ˙
U
T
2 E l
˙
U 2
´
/
` ˙
U
T
2 G l
˙
U 2
´ ,
Q r =
1 + 2
`
˙
u
T
1 E r ˙
u 2
´
/
`
˙
u
T
1 G r ˙
u 2
´
q
1 + 2
`
˙
u
T
1 E r ˙
u 1
´
/
`
˙
u
T
1 G r ˙
u 1
´ q
1 + 2
`
˙
u
T
2 E r ˙
u 2
´
/
`
˙
u
T
2 G r ˙
u 2
´ ,
(1.148)
cos Ψ l =
˙
v
T
1 F
T
r C r F r ˙
v 2
˙
v 1 F T r C r F r ˙
v 2 F T r C r F r
=
c o sΨ r =
˙
V
T
1 F
T
l C l F l ˙
V 2
˙
V 1 F T
l
C l F l
˙
V 2 F T
l
C l F l
=
=
˙
v
T
1 diag(λ
2
1 , λ
2
2 ) ˙
v 2
˙
v 1 D λ ˙
v 2 D λ
,
=
˙
V
T
1 diag(Λ
2
1 , Λ
2
2 ) ˙
V 2
˙
V 1 D Λ ˙
V 2 D Λ
.
(1.149)
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 M
2
l = E
2
A 1 ,A 1 ,A 2
to a sphere M
2
e = S
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 and the right
Cauchy–Green tensor. Second, we derive relative angular shear: Q l = Q r = 1.
End of Example.