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I want to model theinflation of a thin and isotropic circular plastic membrane clamped by a ring. I need to determine the maximum deflection at the pole, stresses, strain, etc..., as a function of the applied pressure difference. The large deflection range complicates it all.

I´m considering plastic materials like this one:DuPont´s Kapton Polyimide Film25 μm (1 mil) type HN . Density 1.42 g/ml, 90 MPa tensile strength at 5% elongation, 231 MPa at break (82% elongation), Poisson ratio 0.34, and tensile modulus 2.5 GPa. (the tensile Stress–Strain curve is attached here).

QUESTION 1:How should I model this kind of membranes?

After spending some days of research, I´m going to briefly share with you my findings.

The problem of inflating a clamped isotropic membrane is known as the classic Hencky´s problem. It has been solved using a Taylor series approach by Hencky [1,2], and reviewed, corrected and improved to include the radial pressure component by Fichter [2] (http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19970023537_1997036944.pdf). In these solutions the knowledge of both the Young modulus and the Poisson ratio leads to the complete determination of the membrane properties. However, in my application, a "close to break" loading is required, so the plastic material should not be modeled as a linear Hookean material, am I right?

Later, I found that Adkin and Rivlin [3] solved the problem in a different way. They analytically determined the manner in which extension ratios and curvatures change at some point. This way, the deformation state at all points can be determined by using numerical integration techniques (Runge-Kutta). The integration can be carried out for any form of the stored-energy function (W). For simplicity, they uses a Mooney material model, where W adopts a simple form:

W = C_{1}(I_{1}-3) + C_{2}(I_{2}-3) (eq.1)

where I_{1}and I_{2}are defined in terms of the principal extension ratios.

Plastics stress-strain curves seem to behave mainly in three different ways (see attached graph): glassy, semi-crystalline or rubber-like. Unfortunately, I´ve read somewhere that Mooney-Rivlin model is only suitable for rubber-like materials.

QUESTION 2:Would I use the Mooney-Rivlin model for non-rubber materials that seem to behave like semi-crystalline polymers above or below Tg?

QUESTION 3:Is there any simpler approach/formula for the approximate modeling of plastic membranes under the large deflection regime?

Thanks a lot in advance for sharing with me your expertise!

Any help will be really appreciated, this is key for me!

REFERENCES:

[1]Hencky, H., “On the stress state in circular plates with vanishing bending stiffness”, Zeitschrift für Mathematik und Physik, Vol. 63, 1915, p. 311-317

[2]FichterW.B. "Some Solutions for the Large Deflections of Uniformly Loaded Circular Membranes", NASA technical paper 3658. (1997) http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19970023537_1997036944.pdf

[3] J. E.Adkinsand R. S.Rivlin. "Large Elastic Deformations of Isotropic Materials. IX. The Deformation of Thin Shells." Phil. Trans. R. Soc. Lond. A 1952 vol. 244 no. 888 505-531

(doi: 10.1098/rsta.1952.0013)

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# Inflation of a clamped isotropic and thin circular plastic membrane

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