 Solution

ID: past-exam-of-the-solid-mechanics-course-of-the-university-of-oxford/2009/3/i/solution

At every point the deformation gradient attains the differentiable energy's minimum, so

D W (D y) = 0

and it is a natural state. By the strict Baker-Ericksen argument just proved, all its principal stretches are equal. In the radial formula this gives

r^{'} (R) = r (R) / R, \frac{d}{d R} (\frac{r ( R )}{R}) = 0.

Thus

r (R) = v R

for one constant

v > 0

on the connected radial interval, and

y (x) = v x .

The constancy is global through the annulus, not a separately chosen dilation at each point.

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