94
1 From Riemann manifolds to Riemann manifolds
Box 1.59 (Left principal stretches).
Left and right matrices of the metric:
G l :=
»
R
2 cos
2 Φ 0
0
R
2
–
,
G r :=
»
1 0
0 1
–
.
(1.348)
Left Jacobi matrix:
J l :=
»
D Λ x D Φ x
D Λ y D Φ y
–
=
= R
2
6
6
4
1
2
(1 + cos Φ) cos Λ/2
− sin Φ sin Λ/2
1
2
(1 + cos Φ) sin β sin Λ/2 cosβ cos Φ + sin β sin Φ cos Λ/2
3
7
7
5 .
(1.349)
Left Cauchy–Green matrix:
C l := J
T
l G r J l =
»
c 11 c 12
c 12 c 22
–
,
c 11 = x
2
Λ + y
2
Λ =
=
1
4
R
2 (1 + cos Φ)
2 `
cos
2 Λ/2 + sin
2 β sin
2 Λ/2
´
=
=
1
4
R
2 (1 + cos Φ)
2 `
1 − cos
2 β sin
2 Λ/2
´
,
c 12 = x Λ x Φ + y Λ y Φ =
=
1
2
R
2 (1 + cos Φ) sin Λ/2 [− sin Φ cos Λ/2 + sin β (cos β cos Φ + sin Φ sin β cos Λ/2)]
=
1
2
R
2 (1 + cos Φ) sin Λ/2
`
sin β cos β cos Φ − sin Φ cos
2 β cos Λ/2
´
=
1
2
R
2 (1 + cos Φ) sin Λ/2 cos β (sin β cos Φ − sin Φ cos β cos Λ/2) ,
c 22 = x
2
Φ + y
2
Φ =
= R
2 ˆ
sin
2 Φ sin
2 Λ/2 + (cos β cos Φ + sin Φ sin β cos Λ/2)
2 ˜
,
(1.350)
det [C l ] = R
4 (1 + cos Φ)
2
4
(sin β sin Φ + cos Φ cos β cos Λ/2)
2 ,
det [G l ] = R
4 cos
2 Φ = det [C l ] .
(1.351)
Left principal stretches:
˛
˛ C l − Λ
2
l G l
˛
˛ = 0 .
(1.352)