Change in tubing volume as a function of pressure

[htsmech]

As I mentioned in another thread, I am looking at a fluidics analogy of an RC circuit to filter out pressure fluctuations. The R is just a constriction, but the C has to be a "balloon", with a relatively linear change in volume vs. pressure. I am thinking of peristaltic tubing, but the walls are very thick (ro/ri=4).

I did a rough calculation for hoop stress in thick walled tubing with the Lame formula, and then tried to estimate volume change based on strain given the stress and modulus.

Has anyone done this before? In particular I am looking for formulas relating volume change to pressure in thick walled tubing.

Thanks,

Steve

 

[Cockroach]

Yes, this is a very common mechanical computation. What you are referring to is diametrical breathing.

I have found Von Mises-Hencky to be the best model, getting excellent agreement between hand computations, FEA and physical laboratory results on turbine meters fitted with magnetic pickup(s). Model your application in 3D and obtain the principle stresses according to your load(s). The Von-Mises-Hencky model applied to thick walled pressure vessels will give you the result:

S = sqrt(3) P [D^2 / (D^2 - d^2)] Von-Mises-Hencky Equation
P = internal pressure
D = outside vessel or pipe diameter
d = internal vessel or pipe diameter
S = normal stress on wall element, 3D

I will spare you the mathematical details regarding hoop, radial and longitudinal stresses associated with thick wall pressure vessels.

Now compute diametrical inflection by Hooke's Law. I have simplified the case noting that longitudinal stress is well pinned, thus fixed. If you need to get fancy, note Poisson's Ratio and figure out wall thinning as a function to expanding inner diameter, like blowing up a balloon.

S = E X eta
E = Young's Modulus
eta = strain on the wall element

The eta variable is were you tinker with the model. I used eta = d' / d = (d2-d1)/d1, i.e. longitundinal axial constrained, set the stresses equal to eachother in order to get an applied pressure. Now you can go backwards and compute volumetric change of a right circular cylinder, simple geometry, and get the corresponding increase in working fluid. It's that simple!

Note that as mentioned, the eta variable is were you can get into the finer points of material mechanics, applying certain principles and building a model particular to your application.

For comparison results, rather than the Von-Mises Hencky Model, a guy to use Beltrami (Maximum Strain Energy) or Saint-Venant and set stresses equal to the Hooke Law part, knock yourself out.

I found Hencky the best though!

Kenneth J Hueston, PEng
Principal
Sturni-Hueston Engineering Inc
Edmonton, Alberta Canada