(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Calculate the deformation of a sphere of radius R and density [itex]\rho[/itex] under the influence of its own gravity. Assume Hooke's law holds for the material.

2. Relevant equations

Not applicable; my question is simply one of understanding.

3. The attempt at a solution

I want to argue that the deformation field should only have a radial component and should only depend on the radial coordinate. Obviously gravity only acts radially and only depends on the radial coordinate. The connection between the deformation and the forces is the stress-strain relationship (Hooke's law), so I need to understand how the components of the stress tensor relate to the forces.

For Cartesian coordinates, I can imagine a cube where each component of the stress tensor has two indices, the first index referring to the face of the cube where the force acts, and the second index referring to the direction of the force.

However, this does not easily extend to a sphere, where the vector normal to the surface is always parallel to the radial direction. Say I wanted to know [itex]\sigma_{\theta r}[/itex]; if the sphere is in equilibrium, then there is no problem because [itex]\sigma_{\theta r}=\sigma_{r \theta}[/itex], but if the sphere is out of equilibrium then I cannot easily compute this component, because I don't understand what it means. Furthermore, I have no way of understanding [itex]\sigma_{\theta \theta}[/itex] or other components not involving [itex]r[/itex].

I would appreciate some kind of mental picture for understanding these components.

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# Homework Help: Stress Tensor in Spherical Coordinates

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