spherically deformed isotropic elastic solid

Say, an infinite isotropic medium with a spherical cavity cut out, and a sphere just a little too big jammed in the void.

I'm trying to set this problem up from scratch, but am mainly scratching my head so far. Obviously we want to come up with a single equation in r or rho or such. Visions of stretched thin spherical shells dance in my head. It should be "simple" because of symmetry.

Any helpful advice or standard reference would be appreciated.

Disclaimer: I am not an engineer, and this is not HW. I think I once set this up correctly, but have since got dumber.

Reply to
Edward Green
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Dear Edward Green:

Google "hoop stress" "spherical shell"

The "inclusion" resolves as an internal pressure...

David A. Smith

Reply to
N:dlzc D:aol T:com (dlzc)

Reply to
BobK207

This is a case of considerable practical interest which has been well presented in standard texts. The concept is classified as a thick wall spherical pressure vessel.

You will see that depending on the fit, the interior surface of the medium may be in plastic yield, or stressed elastically.

Brian W

Reply to
Brian Whatcott

Dear BobK207:

Probably be easier to solve that way, yes. Probably either way, since the inner sphere is "a little too big". Thanks for speaking up.

David A. Smith

Reply to
N:dlzc D:aol T:com (dlzc)

Thanks a million for the search terms and the general advice!

I am primarily interested in the ideal elastic case at present, with stated boundary conditions.

Reply to
Edward Green

All you need are the displacement solutions for the separate interior and exterior problems. The (negative) radial displacement on the surface of a pressurized solid spherical ball of radius R is pR(1-2n)/E, where p is pressure, n is Poisson's ratio, and E is Young's modulus. For a pressurized exterior domain with a spherical cavity of radius R, the radial displacement of the cavity is pR(1+n)/(2E). If the radial mismatch is d (i.e., the ball has an initial radius that exceeds the radius of the cavity by d, a small number), then the sum of the two displacements would be equated to d, since the ball must get a little smaller, and the cavity must get a little bigger.

Reply to
Gordo

Assuming I were only interested in solving for the delta R's. I'm actually interested in the complete exterior solution.

Thank you for those data points though! I have an inkling how to use them to obtain a complete solution, by consistency. Do you happen to have a reference for your formulae?

Reply to
Edward Green

I am not aware of a single book that gives the entire solution other than perhaps Lame's elasticity (1852), but two books discuss the problem, and give useful pieces of the solution: Timoshenko and Goodier, "Theory of Elasticity (McGraw-Hill, 1951), and Sokolnikoff, "Mathematical Theory of Elasticity" (McGraw-Hill, 1956). The latter shows that, for a thick-walled spherical shell, the radial displacement is of the form Ar+B/r^2. (These powers of r are the only ones that satisfy the equilibrium equations.) For the interior problem (inner radius=0), B=0, since displacement must be finite. Similarly, for the exterior problem (outer radius=infinity), A=0. Since stresses are displacement gradients, normal stresses in the radial direction are of the form A+B/r^3 (different constants). The constants can be determined from the B.C. and Hooke's law. When the dust settles, the radial displacement for the exterior problem is (pR(1+n)/(2E))(R/r)^2, and normal stresses in the radial direction are simply -p(R/r)^3.

Reply to
Gordo

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