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Imagine a spherical solid body with initially uniform density, to which we apply a uniform external pressure on its surface. How would the stress distribution look inside that sphere? Would the center of the sphere be more compressed, than its outer part? Or could it also be that the surface of the sphere which is in direct contact with whatever is applying pressure would experience more deformation, but the overall stress distribution inside the body would be uniform?

I would appreciate any intuition or references.

Ruslan Mukhamadiarov
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2 Answers2

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If a solid uniform body (spherical or not) is subjected to a uniform pressure P, then the stress state inside is simply an equitrixial compressive normal stress of magnitude P.

As you note, the relative deformation will be greater with increasing distance from the center of mass.

Chemomechanics
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For a small volume $\Delta V = (\Delta r)(r\Delta\theta)(rsin(\theta))(\Delta\phi) = r^2sin(\theta)\Delta r\Delta\theta \Delta\phi$, the side forces are equal by symmetry. For the radial forces are in equilibrium:$$P_{r+\Delta r}((r+\Delta r)^2)sin(\theta)\Delta\theta \Delta\phi - P_rr^2sin(\theta)\Delta\theta \Delta\phi = 0$$ When $\Delta$'s go to zero, the pressures must be equal. So, for this symmetric situation, the pressure is constant inside the sphere.

Claudio Saspinski
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