3
$\begingroup$

The general form of the cauchy stress tensor is given by the dyadic decomposition

$$\boldsymbol \sigma = \sigma_{ij}\,\,\mathbf{e}_i \otimes \mathbf{e}_j$$

I want to know how this can be expanded in a different coordinate system such as spherical coordinates.

Related: link

$\endgroup$
0
$\begingroup$

You need to express the coordinate basis vectors for the current system in terms of a linear combination of the coordinate basis vectors for spherical coordinates, and substitute into your equation. Alternatively, there is a formula for directly converting the components of your tensor from one coordinate system to another. Both methods give you the same answer.

Method 1: $$\vec{i}_x=\sin{\theta}\cos{\phi}\vec{i}_r+\cos{\theta}\cos{\phi}\vec{i}_{\theta}-sin{\phi}\vec{i}_{\phi}$$ $$\vec{i}_y=\sin{\theta}\sin{\phi}\vec{i}_r+\cos{\theta}\sin{\phi}\vec{i}_{\theta}+cos{\phi}\vec{i}_{\phi}$$ $$\vec{i}_z=\cos{\theta}\vec{i}_r-\sin{\theta}\vec{i}_{\theta}$$

I'm only going to do it for a simple case in which only one component of the stress tensor is non-zero in cartesian coordinates, the z-z component. So,

$$\vec{\sigma}=\sigma_{zz}\vec{i}_z \otimes \vec{i}_z=\sigma_{zz}(\cos{\theta}\vec{i}_r-\sin{\theta}\vec{i}_{\theta})\otimes(\cos{\theta}\vec{i}_r-\sin{\theta}\vec{i}_{\theta})$$ So, $$\vec{\sigma}=\sigma_{zz}\vec{i}_z \otimes \vec{i}_z=\sigma_{zz}(cos^2\theta\vec{i}_r \otimes \vec{i}_r-sin\theta cos\theta(\vec{i}_r \otimes \vec{i}_{\theta}+\vec{i}_{\theta} \otimes \vec{i}_r)+sin^2\theta\vec{i}_{\theta} \otimes \vec{i}_{\theta})$$ So, in this case, it follows that: $$\sigma_{rr}=\sigma_{zz}cos^2\theta$$ $$\sigma_{r\theta}=\sigma_{\theta r}=-\sigma_{zz}sin\theta cos\theta$$ $$\sigma_{\theta \theta}=\sigma_{zz}sin^2\theta$$

$\endgroup$
4
  • $\begingroup$ Done. See above. The remainder of the algebra is up to you. $\endgroup$ Dec 30 '15 at 19:22
  • $\begingroup$ Just substitute, and regroup terms in the dyadics (i.e., algebraically factor). You will also need to express the trigonometric functions of $\theta$ and $\phi $as functions of x, y, and z. $\endgroup$ Dec 30 '15 at 19:37
  • $\begingroup$ I'm failing to do this... I'm barely experienced with dyadics. Can you please derive the result in your answer? $\endgroup$
    – user99917
    Dec 30 '15 at 20:07
  • 1
    $\begingroup$ OK. I showed it for a simple case in which only one component of the stress tensor is non-zero in Cartesian coordinates. $\endgroup$ Dec 30 '15 at 21:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy