Page 246 - Intro to Tensor Calculus

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I 11. Use the results from problem 9 to verify that in cylindrical coordinates

1 ∂(ru θ ) ∂u r
ω θr = −
2r ∂r ∂θ

1 ∂u r ∂u z
ω rz = −
2 ∂z ∂r

1 1 ∂u z ∂u θ
ω zθ = −
2 r ∂θ ∂z
I 12. Use the results from problem 9 to verify that in spherical coordinates

1 ∂(ρu θ ) ∂u ρ
ω θρ = −
2ρ ∂ρ ∂θ

1 1 ∂u ρ ∂(ρu φ)
ω ρφ = −
2ρ sin θ ∂φ ∂ρ

1 ∂(u φ sin θ) ∂u θ
ω φθ = −
2ρ sin θ ∂θ ∂φ
I 13. The conditions for static equilibrium in a linear elastic material are determined from the conservation
law
j
σ i,j + %b i =0, i, j =1, 2, 3,
i
where σ are the stress tensor components, b i are the external body forces per unit mass and % is the density
j
of the material. Assume an orthogonal coordinate system and verify the following results.
(a) Show that
1 ∂
j √ j mj
σ = √ ( gσ ) − [ij, m]σ
i,j j i
g ∂x
(b) Use the substitutions
j h j
σ(ij)= σ no summation on i or j
i
h i
b i
b(i)= no summation on i
h i
ij
σ(ij)= σ h i h j no summation on i or j
and express the equilibrium equations in terms of physical components and verify the relations
3 √ 3 2
X 1 ∂ gh i σ(ij) 1 X σ(jj) ∂(h )
j
√ j − 2 i + h i %b(i)=0,
g ∂x h j 2 h ∂x
j=1 j=1 j
where there is no summation on i.
I 14. Use the results from problem 13 and verify that the equilibrium equations in Cartesian coordinates
can be expressed
∂σ xx ∂σ xy ∂σ xz
+ + + %b x =0
∂x ∂y ∂z
∂σ yx ∂σ yy ∂σ yz
+ + + %b y =0
∂x ∂y ∂z
∂σ zx ∂σ zy ∂σ zz
+ + + %b z =0
∂x ∂y ∂z