Page 283 - Intro to Tensor Calculus

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Figure 2.4-6. Internal pressure on circular hole.

Figure 2.4-7. Tube with internal and external pressure.

I 27. An infinite plane has a circular hole of radius R 1 with an internal pressure P 1 as illustrated in the
figure 2.4-6. Find the stress and displacement fields.
I 28. A tube of inner radius R 1 and outer radius R 0 has an internal pressure of P 1 and an external pressure
of P 0 as illustrated in the figure 2.4-7. Verify the stress and displacement fields derived in example 2.4-7.
I 29. Use Cartesian tensors and combine the equations of equilibrium σ ij,j + %b i =0, Hooke’s law σ ij =
1
λe kk δ ij +2µe ij and the strain tensor e ij = (u i,j + u j,i ) and derive the Navier equations of equilibrium
2
2
∂Θ ∂ u i
σ ij,j + %b i =(λ + µ) + µ + %b i =0,
∂x i ∂x ∂x k
k
where Θ = e 11 + e 22 + e 33 is the dilatation.
I 30. Show the Navier equations in problem 29 can be written in the tensor form

µu i,jj +(λ + µ)u j,ji + %b i =0

or the vector form
~
2
~
µ∇ ~u +(λ + µ)∇ (∇· ~u)+ %b = 0.