Page 280 - Intro to Tensor Calculus

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I 14 . For a linear elastic, homogeneous, isotropic material assume there exists a state of plane strain in
polar coordinates. Verify the equilibrium equations are

∂σ rr 1 ∂σ rθ 1
+ + (σ rr − σ θθ )+ %b r =0
∂r r ∂θ r
2
∂σ rθ 1 ∂σ θθ
+ + σ rθ + %b θ =0
∂r r ∂θ r
∂σ zz
+ %b z =0
∂z
Hint: See problem 15, Exercise 2.3.

I 15. For a linear elastic, homogeneous, isotropic material assume there exists a state of plane stress in
Cartesian coordinates. Verify the equilibrium equations are

∂σ xx ∂σ xy
+ + %b x =0
∂x ∂y
∂σ yx ∂σ yy
+ + %b y =0
∂x ∂y
I 16. Determine the compatibility equations in terms of the Airy stress function φ when there exists a state
of plane stress. Assume the body forces are derivable from a potential function V.

I 17. For a linear elastic, homogeneous, isotropic material assume there exists a state of plane stress in
polar coordinates. Verify the equilibrium equations are

1
∂σ rr 1 ∂σ rθ
+ + (σ rr − σ θθ )+ %b r =0
∂r r ∂θ r
∂σ rθ 1 ∂σ θθ 2
+ + σ rθ + %b θ =0
∂r r ∂θ r