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?

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  • 4
    $\begingroup$ 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. $\endgroup$ – Bill Greene Apr 18 at 9:59
  • $\begingroup$ Also, this question seems to be more on-topic for Physics.SE than Computational Science. $\endgroup$ – 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}


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

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

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