Page 281 - Intro to Tensor Calculus

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I 18. Figure 2.4-4 illustrates the state of equilibrium on an element in polar coordinates assumed to be of
unit length in the z-direction. Verify the stresses given in the figure and then sum the forces in the r and θ
directions to derive the same equilibrium laws developed in the previous exercise.

Figure 2.4-4. Polar element in equilibrium.

Hint: Resolve the stresses into components in the r and θ directions. Use the results that sin dθ ≈ dθ and
2 2
cos dθ ≈ 1 for small values of dθ. Sum forces and then divide by rdr dθ and take the limit as dr → 0and
2
dθ → 0.

I 19. Express each of the physical components of plane stress in polar coordinates, σ rr , σ θθ ,and σ rθ
in terms of the physical components of stress in Cartesian coordinates σ xx , σ yy , σ xy . Hint: Consider the
a
∂x ∂x b
transformation law σ ij = σ ab i j .
∂x ∂x
I 20. Use the results from problem 19 and assume the stresses are derivable from the relations

2
2
2
∂ φ ∂ φ ∂ φ
σ xx = V + , σ xy = − , σ yy = V +
∂y 2 ∂x∂y ∂x 2
where V is a potential function and φ is the Airy stress function. Show that upon changing to polar
coordinates the Airy equations for stress become
2
2
2
1 ∂φ 1 ∂ φ 1 ∂φ 1 ∂ φ ∂ φ
σ rr = V + + 2 2 , σ rθ = 2 − , σ θθ = V + 2 .
r ∂r r ∂θ r ∂θ r ∂r∂θ ∂r
I 21. Verify that the Airy stress equations in polar coordinates, given in problem 20, satisfy the equilibrium
equations in polar coordinates derived in problem 17.