# Confused about one term in the equation for equilibrium of elastic bodies

I am reading this geophysics paper that is trying to explain the deformation of a magma chamber's shell. The basic set up is that we have a sphere at constant pressure $P(t)$ and radius $R1$. We then have a viscoelastic shell with radius ranging from $R1$ to $R2$. This is suspended an in infinite space that is purely elastic $r>R2$.

They begin by just looking at the elastic component of the visco-elastic shell ( viscosity will follow from correspondence principle).They say "according to the equations of equilibrium of elastic bodies" and then list the followeing equation where $u$ is displacement as a function of $r$ is as follow: u(r) = \left\{\begin{aligned} &ar + b/r^2 &&: R1<r<R2\\ &c/r^2 &&: r>R2 \end{aligned} \right.

I don't understand where the $ar$ term comes from in the $R1<r< R2$ region. If we are just looking at the elastic components, why does the equation for that region differ from the region for $r>R2$?

• Welcome on Physics SE :) Can you please add the reference to your question? (And as it is now, I am unsure where the $a$ term comes from too ;) ) Commented Oct 5, 2016 at 15:20
• Haha yeah, that would probably help. Unfortunately I can't find a version to link. The context within the passage references Landau and lifchitz. I looked at their texts, but still have no answer. Commented Oct 5, 2016 at 17:39
• Also, the parper title is Displacement and stress produced by a pressurized, spherical magma chamber, surrounded by a viscoelastic shell Commented Oct 5, 2016 at 17:46
• I think I have an idea. My answer turned out to be a bit long to read, but I hope it helps. Commented Oct 5, 2016 at 20:27

## 1 Answer

For reference, this discussion concerns the paper "Displacement and stress produced by a pressurized, spherical magma chamber, surrounded by a viscoelastic shell" by Michele Dragoni and Cecilia Magnanensi (Physics of the Earth and Planetary Interiors, 56 (1989) 316—328).

Very important in the whole discussion is that the materials are assumed to be compressible, so the $1/r^2$ behaviour has nothing to do with incompressibility (as is often the case in these problems). If we read Landau Lifshitz Vol7: Theory of Elasticity, §7 The equations of equilibrium for isotropic bodies and there look at equation (7.7), we see that the displacement function $\mathbf{u}$ needs to be a biharmonic function, i.e. $$\Delta \Delta \mathbf{u}=0$$ According to A. Oberbeck: Ueber stationäre Flüssigkeitsbewegungen mit Berücksichtigung der inneren Reibung (Journal für die reine und angewandte Mathematik (Crelle's Journal), Band 1876, Heft 81, Seiten 62–80)$^{1}$, the most general solution to this is: $$u=\sum_{n=0}^{\infty}\frac{A_n+B_nr^2}{r^{2n+1}}r^n P_n (\vartheta,\varphi)$$ It is however clear that a subset of the solution needs to be the solutions of $$\Delta \mathbf{u}=0$$ If we either believe Oberbeck or use only the solutions to this Laplace equation and consider the fact that due to the symmetry of the problem, only solutions without dependence on the angles are acceptable solutions, the most general solution for the radial component is: $$u = A r + B r^{-2}$$ For $r > R_2$, the coefficient $A$ needs to equal zero so that the displacement goes to zero as $r \to \infty$. With this, we can now I think understand why this form of the displacement field was assumed as most general form possible.

Physically, Dragoni and Magnanensi write in their paper that your $a \propto \mu_1 - \mu_2$, the difference between the elastic constants of both layers, due to the boundary conditions between the layers. So the $a$ term is important if they have different elastic constants. And this immediately makes sense if we think about the fact that the stress at the surface between the layers needs to be continuous - therefore, in case of a difference in elasticity, the deformation needs to vary throughout the first layer not only by radially decreasing in the expected $r^{-2}$ form but also in another form to guarantee the continuity of the stress across the interlayer boundary.

$^1$: nope, couldn't find a less obscure source

• Quick follow up questions, do you mean that A needs to be zero so that $u = Ar + Br^-2$ goes to zero as r goes to infinity? Commented Oct 11, 2016 at 15:47
• I am happy it was helpful to you and thank you for pointing out my mistake :) Commented Oct 12, 2016 at 11:06
• Hi Sanya, It's been a while since I asked this question, but I was just thinking about it some more today. Do you think there is an analytical limit to how close $R2$ can get to $R1$ and maintain continuity between the surfaces? Commented Nov 1, 2016 at 18:35
• @Chair Sorry for the late reply, life was busy these days. It's a good question and I'm wondering about that. I'll soonish update my answer with a few ideas in that direction but I've got a few job interviews ahead, so please bear with me for a bit longer. Commented Nov 7, 2016 at 15:42
• For my future reference - to be edited into the post with the next edit - sciencedirect.com/science/article/pii/0031920189901660 is the weblink for the source Commented Nov 7, 2016 at 15:42