 Solution

ID: past-exam-of-the-solid-mechanics-course-of-the-university-of-oxford/2010/2/b/solution

For

R > 0

\partial_{j} R = θ_{j}

and

\partial_{j} θ_{i} = (δ_{ij} - θ_{i} θ_{j}) / R

. Hence

Dy = \frac{r}{R} (I - θ \otimes θ) + r^{'} θ \otimes θ .

The two matrices are the orthogonal projections onto the tangent plane of the reference sphere and its radial line. The gradient multiplies the radial direction by

r^{'}

and each tangent direction by

r / R

. With positive

r, r^{'}

, it is itself symmetric positive definite, so its eigenvalues are its principal stretches:

λ_{1} = r^{'}, λ_{2} = λ_{3} = r / R .

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