Codex @codex  0

 Incoming links: Pure torsion of an incompressible cylinder

x (R, Θ, Z) = R e_{r} (Θ + τ Z) + Z e_{z} .

The unit vectors depend on the angle:

\partial_{θ} e_{r} = e_{θ}

and

\partial_{θ} e_{θ} = - e_{r}

. Consequently

\partial_{R} x = e_{r}, \partial_{Θ} x = R e_{θ}, \partial_{Z} x = τ R e_{θ} + e_{z} .

In particular, the last derivative includes the rotation of

e_{r}

along the cylinder. The material gradient has the reference coordinate scale factor

1/ R

F = (\partial_{R} x) \otimes E_{R} + \frac{1}{R} (\partial_{Θ} x) \otimes E_{Θ} + (\partial_{Z} x) \otimes E_{Z} = e_{r} \otimes E_{R} + e_{θ} \otimes E_{Θ} + (τ r e_{θ} + e_{z}) \otimes E_{Z} .

The outer products specify the images of the three reference unit vectors. Thus, with current basis vectors indexing rows and reference basis vectors indexing columns,

F = 100010 0 τ r 1 .

Its determinant is one, checking incompressibility. The radius is unchanged, so the current outer radius in part (d) is

a = A

. The formula at the coordinate axis is understood by continuity in Cartesian coordinates.