Eduardo Kausel-Fundamental solutions in elastodynamics_ a compendium-Cambridge University Press (2006)

18 Fundamentals
with
The dynamic equilibrium equation is
b − ρü + L T σ σ = 0, bT = [b r b θ b z] T = body load vector (1.75)
∂
⎧⎪ ∂ r + 1 r ⎨
Lσ T = 0
⎪ ⎩
− 1 r
1 ∂
r ∂θ
0 0
0 0
0
∂
∂ z
which can be written compactly as
∂
∂ z
1 ∂
r ∂θ
∂
∂ z
0
∂
∂ r + 1 r
⎫
1 ∂
r ∂θ
∂
∂ r + 2 ⎪⎬
r
0
⎪⎭
(1.76)
∂
L σ = L r
∂ r + L 1 ∂
θ
r ∂θ + L ∂
z
∂ z + (L r − L 1 ) 1 (1.77)
r
In full, the operator matrices are
⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫
1 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 1 0
0 0 0
1 0 0
⎪⎨ ⎪⎬ ⎪⎨ ⎪⎬ ⎪⎨ ⎪⎬ ⎪⎨ ⎪⎬
0 0 0
0 0 0
0 0 1
0 0 0
L r =
, L θ =
, L z =
, L 1 =
0 0 0
0 0 1
0 1 0
0 0 0
0 0 1
0 0 0
1 0 0
0 0 0
⎪⎩ ⎪⎭ ⎪⎩ ⎪⎭ ⎪⎩ ⎪⎭ ⎪⎩ ⎪⎭
0 1 0
1 0 0
0 0 0
0 −1 0
(1.78)
Notice that L r ≡ L x , L θ ≡ L y , and L z ≡ L z are the same as in Cartesian coordinates, while
L 1 is new. Also, observe that L σ ≠ L ε .
Stresses in principal surfaces
The stresses in radial, azimuthal and vertical surfaces are
s r = [σ r σ rθ σ rz] T = Lr T σ
∂u
= D rr
∂r + D 1 ∂u
rθ
r ∂θ + D ∂u
rz
∂z + D u
r1
r
s θ = [σ rθ σ θ σ zθ] T = Lθ T σ
∂u
= D θr
∂r + D 1 ∂u
θθ
r ∂θ + D ∂u
θz
∂z + D u
θ1
r
s z = [σ rz σ θz σ z] T = Lz T σ
∂u
= D zr
∂r + D 1 ∂u
zθ
r ∂θ + D ∂u
zz
∂z + D u
z1
r
(1.79)