Young's Modulus and the strain on a sphere due to a uniform pressure

[alex62089]

If I have thin shell like a beach ball inflated with air how would I calculate the change in radius and resistance there of due to the pressure inside the sphere? I have calculated the strain on the sphere to be Pr/2h where P=pressure r=radius h= thickness by cutting the sphere in half and assuming that the pressure over the area pi r^2 = the tension in the shell on this plane (2pir(sigma)) so sigma=Pr/2h. I have tried to use this and various stress strain equations, including the young's modulus equation to calculate the change in radius or surfacearea but so far have failed to do so correctly. My biggest problem is in turing this otherwise linear set of equations into ones that work over an area. Could you please verify my strain equation and help me finish my calculations.

Thank you in advance,
-Alex

 

[Gokul43201]

Your stress equation, \sigma_{\theta} = \sigma _{\phi} = pr/2h is correct.

Next, how did you write out the Hooke's law relation?

And how do you know your answer is wrong?

 

[alex62089]

Stress(sigma)=Young's Modulus(E) * Strain(epsilon or change in length/initial length)

That's what I used and I think that It's wrong because it seems to imply a linear relationship between radial expansion and surface area expansion but intuitively this doesn't seem to be the case. I have tried using dA as the square of dl and differentiating the formula for the surface area of a sphere but don't know if either is mathematically correct.

 

[Gokul43201]

The length you should be using is the circumference.

For small expansions, you will have a nearly linear relationship between linear, suraface and volumetric strains. Recall, 3\alpha = 2\beta = \gamma