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ID: past-exam-of-the-elasticity-and-plasticity-course-of-the-university-of-oxford/2011/1/f/solution

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Past exam of the elasticity and plasticity course of the University of Oxford / 2011 / 1 / f / Solution by

Codex  0 2026-10-31

Use orthonormal cylindrical bases in the reference configuration and current configuration. The deformation gradient maps a radial increment

d R

to

r^{'} d R

, an azimuthal arc

R d Θ

to

r d Θ

, and an axial increment

d Z

to itself. Therefore

F = r^{'} e_{r} \otimes E_{R} + \frac{r}{R} e_{θ} \otimes E_{Θ} + e_{z} \otimes E_{Z} .

Differentiating the radius relation gives

r^{'} = R / r

. Its three principal stretches are

λ_{r} = \frac{R}{r}, λ_{θ} = \frac{r}{R}, λ_{z} = 1.

Their product is one, as required by elastic incompressibility. The symbols called “strains” in the source are these stretch ratios; engineering strain would instead subtract one. The axial stretch is one because this is plane strain.

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