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ID: past-exam-of-the-elasticity-and-plasticity-course-of-the-university-of-oxford/2008/4/b/solution

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Past exam of the elasticity and plasticity course of the University of Oxford / 2008 / 4 / b / Solution by

Codex  0 2026-10-31

For any admissible

v

, put

q = v - w

. At endpoints

q = 0

. On the contact set,

w = f

and

v \geq f

imply

q \geq 0

, while away from contact the reaction

r = p - T w^{''}

is zero. Hence

r q \geq 0

everywhere. Integration by parts gives

T \int_{- 1}^{1} w^{'} (v^{'} - w^{'}) d x + \int_{- 1}^{1} p (v - w) d x = \int_{- 1}^{1} (p - T w^{''}) (v - w) d x = \int_{- 1}^{1} r (v - w) d x \geq 0.

Rearranging proves the requested variational inequality. Piecewise integration is legitimate because the slope is continuous at the contact boundary, so there are no omitted point-force terms.

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