Can gravity cause an expansion of a sphere?

[Jano L.]

Hi everybody,

some time ago our teacher has shown us the following example from the theory of elasticity:

Calculate how the gravity of the sphere changes its size. The sphere is made of ideal linear material (in practice, perhaps some metal) with Young modulus $E$ and Poisson ration $\nu$. The amount of the material is such that if the gravity did not act, the radius of the sphere would be $R_0$. Now imagine the gravity is "turned on". Do you think the sphere will shrink or expand?

Teacher said (and the same can be found in Landau Lifgarbagez, Theory of elasticity, p. 21) that the sphere as a whole will actually expand due to gravity.

Do you think such a strange conclusion can be correct?

 

[A.T.]

Hi everybody,

some time ago our teacher has shown us the following example from the theory of elasticity:

Calculate how the gravity of the sphere changes its size. The sphere is made of ideal linear material (in practice, perhaps some metal) with Young modulus $E$ and Poisson ration $\nu$. The amount of the material is such that if the gravity did not act, the radius of the sphere would be $R_0$. Now imagine the gravity is "turned on". Do you think the sphere will shrink or expand?

Teacher said (and the same can be found in Landau Lifgarbagez, Theory of elasticity, p. 21) that the sphere as a whole will actually expand due to gravity.

Here is that page:

http://books.google.de/books?id=tpY-VkwCkAIC&lpg=PP1&hl=de&pg=PA21#v=onepage&q&f=false

Do you mean problem 12? It is about a spherical cavity in an infinite medium. What sphere do you have in mind? A solid uniform sphere or an empty shell?

 

[Jano L.]

It is the problem 3, p. 21. The problem 12 with the cavity is at the page 24.

 

[A.T.]

It is the problem 3, p. 21

Okay, it is page 19 in the Google version:
http://books.google.de/books?id=tpY-VkwCkAIC&lpg=PP1&hl=de&pg=PA19#v=onepage&q&f=false

Do they actually say that the radius will increase under gravity compared to no gravity? It seems to me they merely say that there is radial compression up to a certain r, and radial stretching above it.

 

[Jano L.]

Aha, I have a second edition. My apologies. I see it is better not to use the page number but rather the paragraph/problem number. Anyway, I can't wait to read what you think of this...

 

[A.T.]

Anyway, I can't wait to read what you think of this...

Well, as I said: I don't see yet that the radius increases under gravity. I guess one would have to integrate the radial strain they give.

 

[Jano L.]

Now I see it, their formula for u implies the sphere as a whole always undergoes compression, so if their solution is correct, there there is no paradox with expansion.

I recall we calculated this in detail and I think we got the result that the sphere expanded. I knew Landau has the same problem and I thought he claims the same thing, but now I see he does not. Most probably we made some mistake.

Thank you for your help,

Jano