Page 282 - Intro to Tensor Calculus

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I 22. In Cartesian coordinates show that the traction boundary conditions, equations (2.3.11), can be
written in terms of the constants λ and µ as

∂u 1 ∂u 1 ∂u 2 ∂u 1 ∂u 3
T 1 = λn 1 e kk + µ 2n 1 1 + n 2 2 + 1 + n 3 3 + 1
∂x ∂x ∂x ∂x ∂x

∂u 2 ∂u 1 ∂u 2 ∂u 2 ∂u 3
T 2 = λn 2 e kk + µ n 1 1 + 2 +2n 2 2 + n 3 3 + 2
∂x ∂x ∂x ∂x ∂x

∂u 3 ∂u 1 ∂u 3 ∂u 2 ∂u 3
T 3 = λn 3 e kk + µ n 1 1 + 3 + n 2 2 + 3 +2n 3 3
∂x ∂x ∂x ∂x ∂x
where (n 1 ,n 2 ,n 3 ) are the direction cosines of the unit normal to the surface, u 1 ,u 2,u 3 are the components
of the displacements and T 1 ,T 2,T 3 are the surface tractions.
I 23. Consider an infinite plane subject to tension in the x−direction only. Assume a state of plane strain
and let σ xx = T with σ xy = σ yy =0. Find the strain components e xx , e yy and e xy . Also find the displacement
field u = u(x, y)and v = v(x, y).

I 24. Consider an infinite plane subject to tension in the y-direction only. Assume a state of plane strain
and let σ yy = T with σ xx = σ xy =0. Find the strain components e xx , e yy and e xy . Also find the displacement
field u = u(x, y)and v = v(x, y).

I 25. Consider an infinite plane subject to tension in both the x and y directions. Assume a state of plane
strain and let σ xx = T , σ yy = T and σ xy =0. Find the strain components e xx ,e yy and e xy . Also find the
displacement field u = u(x, y)and v = v(x, y).

I 26. An infinite cylindrical rod of radius R 0 has an external pressure P 0 as illustrated in figure 2.5-5. Find
the stress and displacement fields.

Figure 2.4-5. External pressure on a rod.