# Determining Displacement Field on a Sphere

I need to find the displacement field for this sphere shape in terms of $$\delta$$. So far, by applying boundary conditions, I know $$u_r = u_\theta = u_\phi = 0$$ at $$r = a$$ and $$u_r = \delta, u_\theta = u_\phi = 0$$ at $$x = b$$.

I am confused on how to go from here, can anyone explain? • Is this a homework problem? I assume you are trying to use the elasticity equations to obtain an analytical solution? You need to show what you have done so far, prior to applying the boundary conditions. – Bill Greene Apr 18 at 9:59
• Also, this question seems to be more on-topic for Physics.SE than Computational Science. – nicoguaro Apr 19 at 15:09

This seems to be more of a Physics question than a Computational Science one.

Due to the symmetry of your problem, you can conclude that the solution is of the form

$$\mathbf{u} = u_r \hat{\mathbf{e}}_r(r)\, ,$$

since the selection of the zenithal and azimuthal angles is arbitrary. This turns the PDE system

$$\begin{equation} (\lambda + \mu) \operatorname{grad} \operatorname{div}\mathbf{u} + \mu \operatorname{rot} \mathbf{u} = \mathbf{0}\, , \end{equation}$$

into

$$\operatorname{grad}\operatorname{div} \mathbf{u} = \mathbf{0}\, ,$$

that turns to be an ODE in $$r$$. After that, you solve your ODE and apply your boundary conditions.