Page 245 - Intro to Tensor Calculus

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I 4. Use the results from problem 1 to write out all components of the strain tensor in spherical coordinates.
Use the notation u(1) = u ρ ,u(2) = u θ ,u(3) = u φ and

e(11) = e ρρ , e(22) = e θθ , e(33) = e φφ , e(12) = e ρθ , e(13) = e ρφ , e(23) = e θφ

to verify the relations

1
∂u ρ 1 ∂u ρ u θ ∂u θ
e ρρ = e ρθ = − +
∂ρ 2 ρ ∂θ ρ ∂ρ

1 ∂u θ u ρ 1 1 ∂u ρ u φ ∂u φ
e θθ = + e ρφ = − +
ρ ∂θ ρ 2 ρ sin θ ∂φ ρ ∂ρ
1 ∂u φ u ρ u θ 1 1 ∂u φ u φ 1 ∂u θ
e φφ = + + cot θ e θφ = − cot θ +
ρ sin θ ∂φ ρ ρ 2 ρ ∂θ ρ ρ sin θ ∂φ
I 5. Expand equation (2.3.67) and find the dilatation in terms of the physical components of an orthogonal
system and verify that

1 ∂(h 2 h 3 u(1)) ∂(h 1 h 3 u(2)) ∂(h 1 h 2 u(3))
Θ= 1 + 2 + 3
h 1 h 2 h 3 ∂x ∂x ∂x
I 6. Verify that the dilatation in Cartesian coordinates is
∂u ∂v ∂w
Θ= e xx + e yy + e zz = + + .
∂x ∂y ∂z

I 7. Verify that the dilatation in cylindrical coordinates is

∂u r 1 ∂u θ 1 ∂u z
Θ= e rr + e θθ + e zz = + + u r + .
∂r r ∂θ r ∂z
I 8. Verify that the dilatation in spherical coordinates is

∂u ρ 1 ∂u θ 2 1 ∂u φ u θ cot θ
Θ= e ρρ + e θθ + e φφ = + + u ρ + + .
∂ρ ρ ∂θ ρ ρ sin θ ∂φ ρ

I 9. Show that in an orthogonal set of coordinates the rotation tensor ω ij can be written in terms of physical
components in the form

1 ∂(h i u(i)) ∂(h j u(j))
ω(ij)= j − i , no summations
2h i h j ∂x ∂x
Hint: See problem 1.
I 10. Use the result from problem 9 to verify that in Cartesian coordinates

1 ∂v ∂u
ω yx = −
2 ∂x ∂y

1 ∂u ∂w
ω xz = −
2 ∂z ∂x

1 ∂w ∂v
ω zy = −
2 ∂y ∂z