40
1 From Riemann manifolds to Riemann manifolds
With these parameterized curves in M
2
l and M
2
r , respectively, we enter Box 1.21. Here, we compute
Φ
−1
l
and Φ
−1
r
in parameterized form, namely (Λ, Φ) → X(Λ, Φ) ∈ R
3 and (λ, φ) → x(λ, φ) ∈ R
3 ,
respectively. The left and the right displacement field is used to derive the tangent vectors { ˙
X 1 , ˙
X 2 } of
type “left” and { ˙
x 1 , ˙
x 2 } of type “right”. The inner products vanish according to our test computations
in Example 1.3. In consequence, Ψ l = Ψ r = π/2 and Σ l = Σ r = 0, i. e. no angular distortion appears.
Box 1.21 (Angular shear or angular distortion).
Left vector field:
Right vector field:
X (Λ, Φ) = E 1
A 1 cos Φ cos Λ
p
1 − E 2 sin
2 Φ
+
x(λ, φ) = e 1 r cos φ cos λ+
+E 2
A 1 cos Φ sin Λ
p
1 − E 2 sin
2 Φ
+ E 3
(1 − E
2 )A 1 sin Φ
p
1 − E 2 sin
2 Φ
.
+e 2 r cos φ sin λ + e 3 r sin φ .
(1.138)
Left displacement field:
Right displacement field:
dX =
∂X
∂Λ
dΛ
dt l
dt l +
∂X
∂Φ
dΦ
dt l
dt l .
dx =
∂x
∂λ
dλ
dt r
dt r +
∂x
∂φ
dφ
dt r
dt r .
(1.139)
1st left parameterized curve:
1st right parameterized curve:
˙
Λ = 1 , ˙
Φ = 0 ,
dX
dt l
=
∂X
∂Λ
.
˙
λ = 1 , ˙
φ = 0 ,
dx
dt r
=
∂x
∂λ
.
(1.140)
2nd left parameterized curve:
2nd right parameterized curve:
˙
Λ = 0 , ˙
Φ = 1 ,
dX
dt l
=
∂X
∂Φ
.
˙
λ = 0 , ˙
φ = 1 ,
dx
dt r
=
∂x
∂φ
.
(1.141)
Left angular shear:
Right angular shear:
˙ ˙
X 1
˛
˛ ˙
X 2
¸
=
fi
∂X
∂Λ
˛
˛
˛
˛
∂X
∂Φ
fl
= 0 ,
0 =
fi
∂x
∂λ
˛
˛
˛
˛
∂x
∂φ
fl
=
˙
˙
x 1
˛
˛ ˙
x 2
¸
,
cos Ψ l = 0 ⇔ Ψ l = ±
π
2
,
±
π
2
= Ψ r ⇔ cos Ψ r = 0 ,
Σ l = Ψ l − Ψ r = 0 .
Σ r = Ψ r − Ψ l = 0 .
(1.142)
1-8 Relative angular shear
A third multiplicative measure of deformation: relative angular shear, Cauchy–Green deformation tensor,
Euler–Lagrange deformation tensor.
The third multiplicative measure of deformation is the ratio Q l and Q r , respectively. This ratio is also
called relative angular shear. In particular
Q l cos Ψ l = cos Ψ r , Q l = Q :=
cos Ψ r
cos Ψ l
versus Q r cos Ψ r = cos Ψ l , Q r = q :=
cos Ψ l
cos Ψ r
, (1.143)
subject to duality Qq = 1. Note that additive angular shear and multiplicative angular shear are
related by
cos Σ l =
c o s Σ r =
versus
= Q l cos
2 Ψ l +
1 − Q 2
l cos 2 Ψ l sin Ψ l
= Q r cos
2 Ψ r +
1 − Q 2
r cos 2 Ψ r sin Ψ r .
(1.144)
1 From Riemann manifolds to Riemann manifolds
With these parameterized curves in M
2
l and M
2
r , respectively, we enter Box 1.21. Here, we compute
Φ
−1
l
and Φ
−1
r
in parameterized form, namely (Λ, Φ) → X(Λ, Φ) ∈ R
3 and (λ, φ) → x(λ, φ) ∈ R
3 ,
respectively. The left and the right displacement field is used to derive the tangent vectors { ˙
X 1 , ˙
X 2 } of
type “left” and { ˙
x 1 , ˙
x 2 } of type “right”. The inner products vanish according to our test computations
in Example 1.3. In consequence, Ψ l = Ψ r = π/2 and Σ l = Σ r = 0, i. e. no angular distortion appears.
Box 1.21 (Angular shear or angular distortion).
Left vector field:
Right vector field:
X (Λ, Φ) = E 1
A 1 cos Φ cos Λ
p
1 − E 2 sin
2 Φ
+
x(λ, φ) = e 1 r cos φ cos λ+
+E 2
A 1 cos Φ sin Λ
p
1 − E 2 sin
2 Φ
+ E 3
(1 − E
2 )A 1 sin Φ
p
1 − E 2 sin
2 Φ
.
+e 2 r cos φ sin λ + e 3 r sin φ .
(1.138)
Left displacement field:
Right displacement field:
dX =
∂X
∂Λ
dΛ
dt l
dt l +
∂X
∂Φ
dΦ
dt l
dt l .
dx =
∂x
∂λ
dλ
dt r
dt r +
∂x
∂φ
dφ
dt r
dt r .
(1.139)
1st left parameterized curve:
1st right parameterized curve:
˙
Λ = 1 , ˙
Φ = 0 ,
dX
dt l
=
∂X
∂Λ
.
˙
λ = 1 , ˙
φ = 0 ,
dx
dt r
=
∂x
∂λ
.
(1.140)
2nd left parameterized curve:
2nd right parameterized curve:
˙
Λ = 0 , ˙
Φ = 1 ,
dX
dt l
=
∂X
∂Φ
.
˙
λ = 0 , ˙
φ = 1 ,
dx
dt r
=
∂x
∂φ
.
(1.141)
Left angular shear:
Right angular shear:
˙ ˙
X 1
˛
˛ ˙
X 2
¸
=
fi
∂X
∂Λ
˛
˛
˛
˛
∂X
∂Φ
fl
= 0 ,
0 =
fi
∂x
∂λ
˛
˛
˛
˛
∂x
∂φ
fl
=
˙
˙
x 1
˛
˛ ˙
x 2
¸
,
cos Ψ l = 0 ⇔ Ψ l = ±
π
2
,
±
π
2
= Ψ r ⇔ cos Ψ r = 0 ,
Σ l = Ψ l − Ψ r = 0 .
Σ r = Ψ r − Ψ l = 0 .
(1.142)
1-8 Relative angular shear
A third multiplicative measure of deformation: relative angular shear, Cauchy–Green deformation tensor,
Euler–Lagrange deformation tensor.
The third multiplicative measure of deformation is the ratio Q l and Q r , respectively. This ratio is also
called relative angular shear. In particular
Q l cos Ψ l = cos Ψ r , Q l = Q :=
cos Ψ r
cos Ψ l
versus Q r cos Ψ r = cos Ψ l , Q r = q :=
cos Ψ l
cos Ψ r
, (1.143)
subject to duality Qq = 1. Note that additive angular shear and multiplicative angular shear are
related by
cos Σ l =
c o s Σ r =
versus
= Q l cos
2 Ψ l +
1 − Q 2
l cos 2 Ψ l sin Ψ l
= Q r cos
2 Ψ r +
1 − Q 2
r cos 2 Ψ r sin Ψ r .
(1.144)