12
1 From Riemann manifolds to Riemann manifolds
f
f
can
f
U l ⊂ M
2
l
U r ⊂ M
2
r
V l ⊂ E
2
V r ⊂ E
2
˜
V l ⊂ E
2
˜
V r ⊂ E
2
Φ l
Φr
˜
Φ l
˜
Φr
Fig. 1.6. Commutative diagram, canonical representation of pairs of metric tensors, “Kartenwechsel” T and
τ , canonical mapping f
can
from the left chart ˜
V l to the right chart ˜
V r .
Box 1.2 (Left versus right Cauchy–Green deformation tensor).
Left CG:
ds
2 = g µν
˘
f
λ (U
L )
¯ ∂u
µ
∂U
M
∂u
ν
∂U
N
dU
M dU
N =
= c MN (U
L ) dU
M dU
N ,
c MN (U
L ) := g µν (U
L )
∂u
µ
∂U M (U
L )
∂u
ν
∂U N (U
L ) .
Right CG:
dS
2 = G MN
˘
F
L (u
λ )
¯ ∂U
M
∂u
µ
∂U
N
∂u
ν du
µ du
ν =
= C µν (u
λ )du
µ du
ν ,
C µν (u
λ ) := G MN (u
λ )
∂U
M
∂u µ (u
λ )
∂U
N
∂u ν (u
λ ) .
(1.49)
Box 1.3 (Left Tissot circle versus left Tissot ellipse, left Cauchy–Green deformation tensor: Ricci calculus).
Left Tissot circle S
1 :
Left Tissot ellipse E
1
λ 1 ,λ 2
:
dS
2 = G MN U
M
A U
N
B
–
dV
A –
dV
B =
d s
2 = g µν u
µ
M u
ν
N U
M
A U
N
B
–
dV
A –
dV
B =
= δ AB –
dV
A –
dV
B =
= Λ
2
1 ( –
dV
1 )
2 + Λ
2
2 ( –
dV
2 )
2 =
= ( –
dV
1 )
2 + ( –
dV
2 )
2 = Ω
2
1 + Ω
2
2 .
= Ω
2
1 /λ
2
1 + Ω
2
2 /λ
2
2 .
(1.50)
Box 1.4 (Left Tissot circle versus left Tissot ellipse, left Cauchy–Green deformation tensor: Cayley calculus).
Left Tissot circle S
1 :
Left Tissot ellipse E
1
λ 1 ,λ 2
:
dS
2 = Ω
T F
T
l G l F l Ω =
d s
2 = Ω
T F
T
l C l F l Ω =
= Ω
T Ω ⇐⇒
= Ω
T diag
`
Λ
2
1 , Λ
2
2
´
Ω ⇐⇒
⇐⇒ F
T
l G l F l = I .
⇐⇒ F
T
l C l F l = diag
`
Λ
2
1 , Λ
2
2
´
= diag
`
1/λ
2
1 , 1/λ
2
2
´
.
(1.51)