 Solution

ID: past-exam-of-the-elasticity-and-plasticity-course-of-the-university-of-oxford/2011/2/g/solution

(u, v)

2 e_{X Y} = u_{Y} + v_{X}

e_{YY} = v_{Y}

, and

tr e = u_{X} + v_{Y}

. The two nontrivial zero-traction conditions become

u_{Y} + v_{X} = 0, λ u_{X} + (λ + 2 μ) v_{Y} = 0 (Y = 0) .

Equivalently for the complex amplitudes they are

U^{'} (0) + ikV (0) = 0

and

ikλ U (0) + (λ + 2 μ) V^{'} (0) = 0

. Because the constitutive equation is real and linear, vanishing of these complex amplitudes enforces the conditions at every position and time.

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