Page 264 - Intro to Tensor Calculus

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and in spherical coordinates

∂ 1 ∂ 2 1 ∂ 1 ∂u φ
(λ + µ) (ρ u ρ )+ (u θ sin θ)+ +
2
∂ρ ρ ∂ρ ρ sin θ ∂θ ρ sin θ ∂φ
2
2 2 ∂u θ 2u θ cot θ 2 ∂u φ ∂ u ρ
2
µ(∇ u ρ − u ρ − − − )+ %b ρ =%
2
2
ρ 2 ρ ∂θ ρ 2 ρ sin θ ∂φ ∂t 2
1 ∂ 1 ∂ 2 1 ∂ 1 ∂u φ
(λ + µ) (ρ u ρ )+ (u θ sin θ)+ +
2
ρ ∂θ ρ ∂ρ ρ sin θ ∂θ ρ sin θ ∂φ
2
2
µ(∇ u θ + 2 ∂u ρ − u θ − 2 cos θ ∂u φ )+ %b θ =% ∂ u θ
2
2
2
2
ρ ∂θ ρ sin θ ρ sin θ ∂φ ∂t 2
2
1 ∂ 1 ∂ 2 1 ∂ 1 ∂u φ
(λ + µ) (ρ u ρ )+ (u θ sin θ)+ +
2
ρ sin θ ∂φ ρ ∂ρ ρ sin θ ∂θ ρ sin θ ∂φ
1 2 2
2 ∂u ρ 2cos θ ∂u θ ∂ u φ
µ(∇ u φ − 2 u φ + 2 + 2 )+ %b φ =% 2
2
2
ρ sin θ ρ sin θ ∂φ ρ sin θ ∂φ ∂t
2
where ∇ is determined from either equation (2.1.12) or (2.1.13).
Boundary Conditions
In elasticity the body forces per unit mass (b i ,i =1, 2, 3) are assumed known. In addition one of the
following type of boundary conditions is usually prescribed:
• The displacements u i , i =1, 2, 3 are prescribed on the boundary of the region R over which a solution
is desired.
• The stresses (surface tractions) are prescribed on the boundary of the region R over which a solution is
desired.
• The displacements u i ,i =1, 2, 3 are given over one portion of the boundary and stresses (surface
tractions) are specified over the remaining portion of the boundary. This type of boundary condition is
known as a mixed boundary condition.
General Solution of Navier’s Equations

There has been derived a general solution to the Navier’s equations. It is known as the Papkovich-Neuber
solution. In the case of a solid in equilibrium one must solve the equilibrium equations

~
2
(λ + µ)∇ (∇· ~u)+ µ∇ ~u + %b =0 or
1 % 1 (2.4.38)
2
∇ ~u + ∇(∇· ~u)+ ~ b =0 (ν 6= )
1 − 2ν µ 2