A Elasticity Equations in Special Coordinate Systems
521
A.3 Spherical Polar Coordinates
The geometry is shown in Fig. A.1.
r = r(R, ,, ,), θ = θ (R, ,, ,), φ = φ(R, ,, ,)
∇ = e R
∂
∂R
+ e
1
R
∂
∂∂
+ e
1
R sin
∂
∂∂
∇ = e r
∂
∂r
+ e θ
1
r
∂
∂θ
+ e φ
1
r sin θ
∂
∂φ
A.3.1 Deformation Gradient
⎡
⎣
F rr F rθ F rφ
F θr F θθ F θφ
F φr F φθ F φφ
⎤
⎦ =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
∂r
∂R
1
R
∂r
∂∂
1
R sin
∂r
∂∂
r
∂θ
∂R
r
R
∂θ
∂∂
r
R sin
∂θ
∂∂
r sin θ
∂φ
∂R
r sin θ
R
∂φ
∂∂
r sin θ
R sin
∂φ
∂∂
⎤
⎥
⎥
⎥
⎥
⎥
⎦
A.3.2 Equations of Motion
∂σ rr
∂r
+
1
r
∂σ θr
∂θ
+
1
r sin θ
∂σ φr
∂φ
+
1
r
2σrr − σ θθ − σ φφ + σ θr cot φ
+ b r = ρa r
∂σ rθ
∂r
+
1
r
∂σ θθ
∂θ
+
1
r sin θ
∂σ φθ
∂φ
+
1
r
3σ rθ + (σ θθ − σ φφ ) cot θ
+ b θ = ρa θ
∂σ rφ
∂r
+
1
r
∂σ θφ
∂θ
+
1
r sin θ
∂σ φφ
∂φ
+
1
r
(3σ rφ + 2σ θφ cot θ) + b φ = ρa φ
Fig. A.1 Spherical polar
coordinate system
I
T
r
x
y
z
e r
e I
e T