1-2 Stretch or length distortion 13
Box 1.5 (The right Tissot ellipse versus the right Tissot circle, right Cauchy–Green deformation tensor:
Ricci calculus).
Right Tissot ellipse E
1
Λ 1 ,Λ 2 :
Right Tissot circle S
1 :
dS
2 = G MN U
M
µ U
N
ν u
µ
α u
ν
β
–
dv
α –
dv
β =
d s
2 = g µν u
µ
α u
ν
β
–
dv
α –
dv
β =
= ( –
dv
1 )
2 /Λ
2
1 + ( –
dv
2 )
2 /Λ
2
2 =
= δ αβ –
dv
α –
dv
β =
= λ
2
1 ω
2
1 + λ
2
2 ω
2
2 .
= ( –
dv
1 )
2 + ( –
dv
2 )
2 = ω
2
1 + ω
2
2 .
(1.52)
Box 1.6 (The right Tissot ellipse versus the right Tissot circle, right Cauchy–Green deformation tensor:
Cayley calculus).
Right Tissot ellipse E
1
Λ 1 ,Λ 2 :
Right Tissot circle S
1 :
dS
2 = ω
T F
T
r C r F r ω =
d s
2 = ω
T F
T
r G r F r ω =
= ω
T diag
`
λ
2
1 , λ
2
2
´
ω
= ω
T ω
⇐⇒
⇐⇒
F
T
r C r F r = diag
`
λ
2
1 , λ
2
2
´
= diag
`
1/Λ
2
1 , 1/Λ
2
2
´
.
F
T
r G r F r = I .
(1.53)
Box 1.7 (Left general eigenvalue problem and right general eigenvalue problem: Ricci calculus).
Left eigenvalue problem:
Right eigenvalue problem:
Λ
2 dS
2 = ds
2 ,
λ
2 ds
2 = dS
2 ,
Λ
2 G MN U
M
A U
N
B
–
dV
A –
dV
B =
λ
2 g µν u
µ
α u
ν
β
–
dv
α –
dv
β =
= g µν u
µ
M u
ν
N U
M
A U
N
B
–
dV
A –
dV
B
= G MN U
M
µ U
N
ν u
µ
α u
ν
β
–
dv
α –
dv
β
⇐⇒
⇐⇒
Λ
2 G MN U
N
B = c MN U
N
B
λ
2 g µν u
ν
β = C µν u
ν
β
⇐⇒
⇐⇒
(c MN − Λ
2 G MN )U
N
B = 0 ,
(C µν − λ
2 g µν )u
ν
β = 0 ,
subject to
subject to
g µν u
µ
M u
ν
N U
M
A U
N
B = δ AB .
G MN U
M
µ U
N
ν u
µ
α u
ν
β = δ µν .
(1.54)
Box 1.8 (Left general eigenvalue problem and right general eigenvalue problem: Cayley calculus).
Left eigenvalue problem:
Right eigenvalue problem:
Λ
2 dS
2 = ds
2 ,
λ
2 ds
2 = dS
2 ,
Λ
2 –
dV
T F
T
l G l F l
–
dV =
λ
2 –
dV
T F
T
r G r F r –
dV =
= –
dV
T F
T
l C l F l
–
dV
= –
dV
T F
T
r C r F r –
dV
⇐⇒
⇐⇒
(C l − Λ
2 G l )F l = 0 ,
(C r − λ
2 G r )F r = 0 ,
subject to
subject to
F
T
l G l F l = I .
F
T
r G r F r = I .
(1.55)
Box 1.5 (The right Tissot ellipse versus the right Tissot circle, right Cauchy–Green deformation tensor:
Ricci calculus).
Right Tissot ellipse E
1
Λ 1 ,Λ 2 :
Right Tissot circle S
1 :
dS
2 = G MN U
M
µ U
N
ν u
µ
α u
ν
β
–
dv
α –
dv
β =
d s
2 = g µν u
µ
α u
ν
β
–
dv
α –
dv
β =
= ( –
dv
1 )
2 /Λ
2
1 + ( –
dv
2 )
2 /Λ
2
2 =
= δ αβ –
dv
α –
dv
β =
= λ
2
1 ω
2
1 + λ
2
2 ω
2
2 .
= ( –
dv
1 )
2 + ( –
dv
2 )
2 = ω
2
1 + ω
2
2 .
(1.52)
Box 1.6 (The right Tissot ellipse versus the right Tissot circle, right Cauchy–Green deformation tensor:
Cayley calculus).
Right Tissot ellipse E
1
Λ 1 ,Λ 2 :
Right Tissot circle S
1 :
dS
2 = ω
T F
T
r C r F r ω =
d s
2 = ω
T F
T
r G r F r ω =
= ω
T diag
`
λ
2
1 , λ
2
2
´
ω
= ω
T ω
⇐⇒
⇐⇒
F
T
r C r F r = diag
`
λ
2
1 , λ
2
2
´
= diag
`
1/Λ
2
1 , 1/Λ
2
2
´
.
F
T
r G r F r = I .
(1.53)
Box 1.7 (Left general eigenvalue problem and right general eigenvalue problem: Ricci calculus).
Left eigenvalue problem:
Right eigenvalue problem:
Λ
2 dS
2 = ds
2 ,
λ
2 ds
2 = dS
2 ,
Λ
2 G MN U
M
A U
N
B
–
dV
A –
dV
B =
λ
2 g µν u
µ
α u
ν
β
–
dv
α –
dv
β =
= g µν u
µ
M u
ν
N U
M
A U
N
B
–
dV
A –
dV
B
= G MN U
M
µ U
N
ν u
µ
α u
ν
β
–
dv
α –
dv
β
⇐⇒
⇐⇒
Λ
2 G MN U
N
B = c MN U
N
B
λ
2 g µν u
ν
β = C µν u
ν
β
⇐⇒
⇐⇒
(c MN − Λ
2 G MN )U
N
B = 0 ,
(C µν − λ
2 g µν )u
ν
β = 0 ,
subject to
subject to
g µν u
µ
M u
ν
N U
M
A U
N
B = δ AB .
G MN U
M
µ U
N
ν u
µ
α u
ν
β = δ µν .
(1.54)
Box 1.8 (Left general eigenvalue problem and right general eigenvalue problem: Cayley calculus).
Left eigenvalue problem:
Right eigenvalue problem:
Λ
2 dS
2 = ds
2 ,
λ
2 ds
2 = dS
2 ,
Λ
2 –
dV
T F
T
l G l F l
–
dV =
λ
2 –
dV
T F
T
r G r F r –
dV =
= –
dV
T F
T
l C l F l
–
dV
= –
dV
T F
T
r C r F r –
dV
⇐⇒
⇐⇒
(C l − Λ
2 G l )F l = 0 ,
(C r − λ
2 G r )F r = 0 ,
subject to
subject to
F
T
l G l F l = I .
F
T
r G r F r = I .
(1.55)