Page 247 - Intro to Tensor Calculus

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I 15. Use the results from problem 13 and verify that the equilibrium equations in cylindrical coordinates
can be expressed
∂σ rr 1 ∂σ rθ ∂σ rz 1
+ + + (σ rr − σ θθ )+ %b r =0
∂r r ∂θ ∂z r
2
∂σ θr 1 ∂σ θθ ∂σ θz
+ + + σ θr + %b θ =0
∂r r ∂θ ∂z r
∂σ zr 1 ∂σ zθ ∂σ zz 1
+ + + σ zr + %b z =0
∂r r ∂θ ∂z r
I 16. Use the results from problem 13 and verify that the equilibrium equations in spherical coordinates
can be expressed

∂σ ρρ 1 ∂σ ρθ 1 ∂σ ρφ 1
+ + + (2σ ρρ − σ θθ − σ φφ + σ ρθ cot θ)+ %b ρ =0
∂ρ ρ ∂θ ρ sin θ ∂φ ρ
1 1
∂σ θρ 1 ∂σ θθ ∂σ θφ
+ + + (3σ ρθ +[σ θθ − σ φφ ]cot θ)+ %b θ =0
∂ρ ρ ∂θ ρ sin θ ∂φ ρ
∂σ φρ 1 ∂σ φθ 1 ∂σ φφ 1
+ + + (3σ ρφ +2σ θφ cot θ)+ %b φ =0
∂ρ ρ ∂θ ρ sin θ ∂φ ρ

I 17. Derive the result for the Lagrangian strain defined by the equation (2.3.60).
I 18. Derive the result for the Eulerian strain defined by equation (2.3.61).

i
j
i
I 19. The equation δa = u a , describes the deformation in an elastic solid subjected to forces. The
,j
i
i
i
quantity δa denotes the difference vector A − a between the undeformed and deformed states.
i
i
(a) Let |a| denote the magnitude of the vector a and show that the strain e in the direction a can be
represented
i j
δ|a| a a i j
e = = e ij = e ij λ λ ,
|a| |a| |a|
i
i
where λ is a unit vector in the direction a .
2
3
i
1
(b) Show that for λ =1,λ =0,λ = 0 there results e = e 11 , with similar results applying to vectors λ in
the y and z directions.
2
i j
Hint: Consider the magnitude squared |a| = g ij a a .
2
I 20. At the point (1, 2, 3) of an elastic solid construct the small vector ~a = ( ˆ e 1 + 2 ˆ e 2 + 1 ˆ e 3 ), where
3 3 3
> 0 is a small positive quantity. The solid is subjected to forces such that the following displacement field
results.
−2
~u =(xy ˆ e 1 + yz ˆ e 2 + xz ˆ e 3 ) × 10
~
Calculate the deformed vector A after the displacement field has been imposed.
I 21. For the displacement field
2 2
~u =(x + yz) ˆ e 1 +(xy + z ) ˆ e 2 + xyz ˆ e 3
(a) Calculate the strain matrix at the point (1, 2, 3).
(b) Calculate the rotation matrix at the point (1, 2, 3).