Stress Tensor in Spherical Coordinates

[McCoy13]

Homework Statement

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

Homework Equations

Not applicable; my question is simply one of understanding.

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 $\sigma_{\theta r}$; if the sphere is in equilibrium, then there is no problem because $\sigma_{\theta r}=\sigma_{r \theta}$, 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 $\sigma_{\theta \theta}$ or other components not involving $r$.

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

 

[Chestermiller]

Google Hooke's Law in Spherical Coordinates.

You also need to know the strain-displacement relationships in spherical coordinates. But you should be aware that, for your problem, the principal directions of stress and strain are in the radial, latitudinal, and longitudinal directions. There are no shear components of stress and strain. It also helps to assume that you have a sphere in which the stresses and strains are zero until after gravity is suddenly "switched on," and a new equilibrium is established.

Start out with the stress-equilibrium equations in spherical coordinates. Recognize the symmetries of the problem, so that the radial displacement is the only non-zero displacement, and derivatives with respect to latitude and longitude are zero.

First express the gravitational acceleration as a function of radial position within the sphere. Then write down the strain-displacement equations in spherical coordinates, under the symmetrical constraints. Then substitute these equations into the Hooke's law equations. Then substitute these equations, together with acceleration of gravity expression into the stress-equilibrium equations. Can you figure out what the boundary condition on stress is at the outer radius?

This procedure should provide an ordinary differential equation for the radial displacement.