Paper bag problem for irregular shapes 2

[Davide Recchia]

How would I calculate the maximum possible inflated volume of an initially flat sealed bag which has an irregular shape (two examples below) made out of two pieces of material which can bend but not stretch and assuming the opening is on the top edge?

For a rectangular shape, this is commonly referred to as the paper bag problem:

https://en.wikipedia.org/wiki/Paper_bag_problem

 

[3DDave]

Make a bag and see. Since the paper bag problem seems to be predicated on a uniform internal pressure, having this open at the top doesn't fit that, but I suppose it can be approximated by adding a long rectangular section to the open edge that would inflate to a cylinder.

The main problem is that the Gauss curvature for the flat item is 0 and, if the material is considered rigid in tension, that number will remain 0, which is incompatible with the two directional bending that is necessary for these shapes to inflate. This discrepancy is sorted out in actual items by a combination of stretch of the material and the formation of creases with negative curvature.

(fixed Euler -> Gauss)

 

[TheTick]

Max possible volume for a given surface area is a sphere. Calculate the sphere volume for the area, and you'll know for sure you have less than that.

 

[GregLocock]

I'd sort out the open end question by developing a solution for a mirror image of the thing connected at the open edge.

But I haven't got the faintest idea how to do that,

Taking tick's idea further, the maximum volume for a given flat shape would be circular at each cross section. So that gives a different, lower, upper estimate for the volume.

Cheers

Greg Locock


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[BUGGAR]

the "inflated membrane" theory is used to determine shear (deflection) in structural shapes. Just use that.

 

[GregLocock]

Good hint, with wrinkles!

https://www.sciencedirect.com/science/article/pii/S0020768308002412

This paper proposes an effective method for directly determining the final equilibrium shapes of closed inextensible membranes subjected to internal pressures. With reference to new high-performance textile materials, we assume that the mechanical response of a fabric membrane can be accurately represented by regarding it as a two-state material. In the active state, the membrane is subject to tensile stresses and is virtually inextensible; vice versa, in the passive state it is unable to sustain any compressive stress, so it contracts freely. Equilibrium of the membrane in the final configuration is enforced by recourse to the minimum total potential energy principle. The Lagrange multipliers method is used to solve the minimum problem by accounting for the aforesaid nonlinear constitutive law. The set of governing equations is solved for the unknown coordinates of the equilibrium surface points. Closed form solutions are obtained for fully wrinkled cylindrical and axisymmetric membranes under homogeneous boundary conditions, while a simple iterative procedure is used to numerically solve cases of axisymmetric membranes under various inhomogeneous boundary conditions. The soundness of the proposed method is verified by comparing the results with solutions available in the literature.

Cheers

Greg Locock


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[chicopee]

If the bag is water resistant I would measure the amount of water that it can hold.

 

[GregLocock]

Eureka!

Cheers

Greg Locock


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[Davide Recchia]

Thank you all for the comments.

Measuring the volume is not a problem. I've made a template, cut out the shape and sealed the edges.
I was just wondering if there was a faster method (I really don't have time to understand the math behind it)
to generate shapes given a volume as input.

So far, I've made the smallest and largest bags, measured the volume, and interpolated the areas to derive sizes in between.

 

[IRstuff]

I've seen a couple of websites tout their ability to compute an answer to that specific question, but it seemed to be all proprietary, although one site seemed to be using FEA for that.

TTFN (ta ta for now)
I can do absolutely anything. I'm an expert!

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[BUGGAR]

"(I really don't have time to understand the math behind it)
to generate shapes given a volume as input."

Nor do I.

 

[Davide Recchia]

I did try a simulation in ANSYS.

To find out the volume, I modeled the bag as an open shell and applied pressure on the inner sides.

Then exported the deformed shape as STL file, capped the opening to get a volume in Solidworks.

 

[btrueblood]

For a shell analysis to work, you need to consider large deflections and large bending angles at the joins...dunno if that works well.

 

[Davide Recchia]

Works reasonably well