520
A Elasticity Equations in Special Coordinate Systems
A.1.2 Equations of Motion
∂σ xx
∂x
+
∂σ yx
∂y
+
∂σ zx
∂z
+ b x = ρa x
∂σ xy
∂x
+
∂σ yy
∂y
+
∂σ zy
∂z
+ b y = ρa y
∂σ xz
∂x
+
∂σ yz
∂y
+
∂σ zz
∂z
+ b z = ρa z
A.2 Cylindrical Polar Coordinates
r = r(R, ,, Z), θ = θ(R, ,, Z), z = z(R, ,, Z)
∇ = e R
∂
∂R
+
e
R
∂
∂∂
+ e Z
∂
∂Z
∇ = e r
∂
∂r
+
e θ
r
∂
∂θ
+ e z
∂
∂z
A.2.1 Deformation Gradient
⎡
⎣
F rr F rθ F rz
F θr F θθ F θz
F zr F zθ F zz
⎤
⎦ =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
∂r
∂R
1
R
∂r
∂∂
∂r
∂Z
r
∂θ
∂R
r
R
∂θ
∂∂
r
∂θ
∂Z
∂z
∂R
1
R
∂z
∂∂
∂z
∂Z
⎤
⎥
⎥
⎥
⎥
⎥
⎦
A.2.2 Equations of Motion
∂σ rr
∂r
+
1
r
∂σ θr
∂θ
+
∂σ zr
∂z
+
1
r
(σ rr − σ θθ ) + b r = ρa r
∂σ rθ
∂r
+
1
r
∂σ θθ
∂θ
+
∂σ zθ
∂z
+
2σ rθ
r
+ b θ = ρa θ
∂σ rz
∂r
+
1
r
∂σ θz
∂θ
+
∂σ zz
∂z
+
σ rz
r
+ b z = ρa z