90
1 From Riemann manifolds to Riemann manifolds
Box 1.56 (The canonical representation of left angular shear and right angular shear. Special case: G 12 = 0,
c 12 = 0 and g 12 = 0, C 12 = 0).
Left and right angular parameters Ψ l and Ψ r :
cos Ψ l =
√
C 11
du
ds
ds
dS
versus
√ c 11
dU
dS
dS
ds
= cos Ψ r ,
cos Ψ l =
√
C 11
du
ds
1
λ
versus
√ c 11
dU
dS
1
Λ
= cos Ψ r ,
sin Ψ l =
√
C 22
dv
ds
ds
dS
versus
√
c 22
dV
dS
dS
ds
= sin Ψ r ,
sin Ψ l =
√
C 22
dv
ds
1
λ
versus
√
c 22
dV
dS
1
Λ
= sin Ψ r .
(1.331)
Left and right stretches, left and right principal stretches:
λ
2 :=
dS
2
ds 2
versus Λ
2 :=
ds
2
dS 2 ,
λ
2
1 =
C 11
g 11
versus Λ
2
1 =
c 11
G 11
,
λ
2
2 =
C 22
g 22
versus Λ
2
2 =
c 22
G 22
,
(1.332)
cos Ψ r =
√ g 11 u
√
G 11 U
= cos Ψ l
⇒
⇒
cos Ψ l = cos Ψ r
r
C 11
g 11
1
λ
versus cos Ψ l
r
c 11
G 11
1
Λ
= cos Ψ r ,
sin Ψ l = sin Ψ r
r
C 22
g 22
1
λ
versus sin Ψ l
r
c 22
G 22
1
Λ
= sin Ψ r ,
(1.333)
cos Ψ l = cos Ψ r
λ 1
λ
versus cos Ψ l
Λ 1
Λ
= cos Ψ r ,
sin Ψ l = sin Ψ r
λ 2
λ
versus
sin Ψ l
Λ 2
Λ
= sin Ψ r ,
tan Ψ l =
λ 2
λ 1
tan Ψ r versus
Λ 2
Λ 1
tan Ψ l = tan Ψ r .
(1.334)
Left and right angular shear, left and right angular distortion:
tan (Ψ l − Ψ r ) = tan
P
l
versus
tan
P
r = tan (Ψ r − Ψ l ) ,
tan
P
l =
tan Ψ l − tan Ψ r
1 + tan Ψ l tan Ψ r
versus
tan
P
r =
tan Ψ r − tan Ψ l
1 + tan Ψ r tan Ψ l
,
tan
P
l =
tan Ψ l − Λ 2 Λ
−1
1 tan Ψ l
1 + Λ 2 Λ
−1
1 tan
2 Ψ l
versus
tan
P
r =
tan Ψ r − λ 2 λ
−1
1 tan Ψ r
1 + λ 2 λ
−1
1 tan
2 Ψ r
,
tan
P
l = (Λ 1 − Λ 2 )
tan Ψ l
Λ 1 + Λ 2 tan
2 Ψ l
versus tan
P
r =
tan Ψ r
λ 1 + λ 2 tan
2 Ψ r
(λ 1 − λ 2 ) .
(1.335)