Inflation of a clamped isotropic and thin circular plastic membrane

[Vigardo]

Appreciated experts,

I want to model the inflation 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 Film 25 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 = C₁(I₁-3) + C₂(I₂-3) (eq.1)

where I₁ and I₂ 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] Fichter W.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. Adkins and 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)

 

[Achy47]

Thank you for sharing your research and findings on modeling the inflation of a thin and isotropic circular plastic membrane. As a scientist with expertise in material properties and mechanics, I would like to offer some insights and suggestions for your study.

Firstly, your choice of DuPont's Kapton Polyimide Film 25 μm (1 mil) type HN as the material for your membrane is suitable for its high tensile strength and low density. However, as you have mentioned, the large deflection range complicates the analysis. In order to accurately model the behavior of the membrane under these conditions, it is important to consider the material's non-linear behavior and the effects of large deformations.

To address your first question, there are several approaches to modeling this type of membrane. One option is to use the classical Hencky's solution, which involves a Taylor series approach and accounts for the radial pressure component. However, as you have mentioned, this solution assumes a linear Hookean material behavior and may not accurately capture the behavior of plastic materials under close to break loading conditions.

Another approach, as you have mentioned, is the Adkin and Rivlin solution which uses a Mooney material model to determine the deformation state at all points on the membrane. This approach allows for non-linear material behavior and can be applied to a variety of stored-energy functions, including the Mooney-Rivlin model. However, as you have also pointed out, this model is more suitable for rubber-like materials and may not accurately capture the behavior of semi-crystalline polymers.

To address your second question, it is possible to use the Mooney-Rivlin model for non-rubber materials, but the accuracy of the results may be limited. It is important to consider the material's stress-strain behavior and choose a suitable stored-energy function that best represents the behavior of the material.

In terms of simplifying the modeling approach, there are some approximate methods that can be used for large deflection analysis of membranes. One option is the Geometrically Nonlinear Membrane Theory (GNMT), which uses a simplified approach to determine the maximum deflection and stresses in the membrane. However, this approach may not accurately capture the behavior of plastic materials and may require additional modifications to account for non-linear behavior.

In conclusion, there is no one-size-fits-all approach to modeling the inflation of a thin and isotropic circular plastic membrane. It is important to carefully consider