Can plastic materials be modeled as Mooney-Rivlin materials?

[Vigardo]

Dear experts,

I wonder if semi-crystalline plastics can be considered as Mooney-Rivlin materials for an application in the analytic modeling of thin membranes. Any help will be very appreciated!

The stored-energy function (W) of a Mooney material model is:

W = C₁(I₁-3) + C₂(I₂-3)

Where I₁ and I₂ are defined in terms of the principal extension ratios (λ):

I₁ = λ₁² + λ₂² + λ₃²
I₂ = λ₁² λ₂² + λ₂² λ₃² + λ₃² λ₁²

I´m considering plastic materials like this one:
- DuPont´s Kapton Polyimide Film 25 μm (1 mil) type HN .
Density: 1.42 g/ml
Tensile strength at 5% elongation: 90 MPa
tensile strength at break: 231 MPa (82% elongation)
Poisson ratio: 0.34
Tensile modulus: 2.5 GPa

(the tensile Stress–Strain curve is attached in this post).

 

[afreiden]

So, beyond 5% strain, your material is no longer elastic? Your plots are showing 70% strain, so I assume that is mostly plastic behavior. The Mooney Rivlin model is an elastic model.

Are you asking if you can simulate the response of some structure composed of this material (thin membrane, as you say) using a Mooney-Rivlin model that is characterized from the tensile-test data that you attached? I don't think that would be accurate at all. You need to use some kind of damage model.

You might consider something simpler, like a steel model with plasticity. I would bet that such a model would give better results than any hyperelastic model, if you are well beyond the elastic limit of your material.

 

[Chestermiller]

What kind of deformation is this thin film experiencing (kinematically)? Is this in a process, or in a product application? Is the crystallinity changing during the deformation? Is the temperature changing during the deformation?

Chet

 

[Vigardo]

First, thank you very much for your answers!

Afreiden:

After posting I found that yield strain for Kapton HN is around 3%, so as you´ve pointed out, it is not within the elastic range at all. Sorry, I wrongly thought that the material would go back to its initial shape after such a large strain (≈70%).

Would you provide me more information about "some kind of damage model" or a "steel model with plasticity" that is suitable for a thin plastic membrane under such large strains?

Chestermiller:

What kind of deformation is this thin film experiencing (kinematically)? Is this in a process, or in a product application?

Sorry for my ignorance, but I don´t understand very well what you mean with "kinematically". If you meant that if it´s experiencing a dynamic or static loading, it is static. The membrane I´m trying to model belongs to a device to store pressurized air. It is intended to withstand around 100 kPa of constant pressure differential between both sides. The project is at a very initial design stage, all I´m talking about is theoretical at this moment.

Is the crystallinity changing during the deformation? Is the temperature changing during the deformation?
Chet

An approximately constant ambient temperature (25 ºC) is considered in my study. Since it does not change, crystallinity shouldn´t be a matter, should it?

Finally, I wanted to say that any formula, link or reference of book or paper, would be highly appreciated!

Thanks again!

 

[Chestermiller]

First, thank you very much for your answers!

Chestermiller:
It is intended to withstand around 100 kPa of constant pressure differential between both sides. The project is at a very initial design stage, all I´m talking about is theoretical at this moment.

An approximately constant ambient temperature (25 ºC) is considered in my study. Since it does not change, crystallinity shouldn´t be a matter, should it?

Finally, I wanted to say that any formula, link or reference of book or paper, would be highly appreciated!

Thanks again!

OK. So you have a film membrane as part of a device, and it's supporting a pressure differential of 100kPa. What is the geometry and the dimensions. For example, is it circular, and held down at its edges. What is the diameter. How much strain do you expect the film to experience (radially)? What is the thickness of the film? Have you done any modelling calculations to determine the deformation if the stress-strain behavior were in the linear region? If so, what strains did you estimate? Do you expect the film to creep under load?

Chet

 

[Vigardo]

OK. So you have a film membrane as part of a device, and it's supporting a pressure differential of 100kPa. What is the geometry and the dimensions. For example, is it circular, and held down at its edges. What is the diameter. How much strain do you expect the film to experience (radially)? What is the thickness of the film?

Basically the model should correspond to the classic Hencky problem [1]. 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.

I don´t know yet the exact dimensions, but the diameter would be from 0.1 to 1 m and the thickness in the 100 μm range. It should be working in the large deflection range because, in principle, I want to use all available internal strength of the material. I suppose this complicates it all.

Have you done any modelling calculations to determine the deformation if the stress-strain behavior were in the linear region? If so, what strains did you estimate? Do you expect the film to creep under load?

I found a revised version of Hencky´s formulas in [2]. Assuming an elastic behavior I found a dimension-less deflection at the pole (w0) around 0.2 or 0.3. I did not integrate the full strain along the membrane but it seems far away the 3% of the elastic limit of Kapton films or other similar high strength plastics. In principle, I would not consider creep.

By the way, after comparing [1] with [2], I detected a mistake (perhaps an typo) in the equation 12 of [2] for the the radial and circunfererntial stresses. I think the coefficient before the corresponding summations is wrong. The (1/4)q^2/3 in [2] should be substituted by (1/4)(E(pa/h)²)^1/3.


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/ca...1997036944.pdf

 

[Vigardo]

Sorry, I mistyped the link to the reference [2]. Here is the correct one:

[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.pdf

 

[Chestermiller]

Hi. I looked over your reference (briefly), and I have some ideas on this. If you were only interested in the maximum deflection, and not the stresses and strains, you could probably use your device, together with dimensionless analysis to establish this relation for a wide range of conditions with only a limited number of experiments. You could do more if you measured the full deformational kinematics experimentally, again using dimensional analysis.

This is a very interesting deformational mechanics problem, involving large displacements and possibly large strains. Your reference seems to take into account the large displacement aspect of the problem. The principal directions of stress and strain are going to line up with the meridional and circumferential directions. Unfortunately, the circumferential strain at the support is going to be zero. I'm wondering what the strains vs initial meridional position look like, and what the strains plotted against one another look like. This might suggest some biaxial stretching experiments that could be used to customize the measurements of stress strain behavior to the deformation kinematics in the actual system. This would add accuracy to the model analysis. I feel that, in a specific and well-defined deformation regime like this, this type of approach would be preferable to using a general rheological model (for which the material parameters are measured in experiments very different from the kinematics experienced in the actual device).

You do also realize that, if you exceed the yield strain, the membrane is not going to return to its initial configuration after you remove the pressure load?

Chet

 

[Vigardo]

Hi. I looked over your reference (briefly), and I have some ideas on this. If you were only interested in the maximum deflection, and not the stresses and strains, you could probably use your device, together with dimensionless analysis to establish this relation for a wide range of conditions with only a limited number of experiments. You could do more if you measured the full deformational kinematics experimentally, again using dimensional analysis.

At this moment my project (device) is purely theoretic. It only will become practical if it seems feashible "on paper". So any experiment is discarded by now.

By the way, besides maximum deflection, I need to model stresses and strain to determine the thickness that minimizes the necessary amount of film.

... I'm wondering what the strains vs initial meridional position look like, and what the strains plotted against one another look like.

I used Henky´s equations to compute and plot circunferential and meridional stresses and strains. I will share with you the plots in my next post. (I´m not at my computer know)

You do also realize that, if you exceed the yield strain, the membrane is not going to return to its initial configuration after you remove the pressure load?

Yes, now I do. But in principle it should not be a problem as long as the ultimate (or design) film strength is maintained.

------------------------------------------------------------------------------------------

The problem of using Henky´s formulation (and related ones: Fischer´s [2]) is that it considers the material as Hookean, but the stress-strain curve shown above is far from it. Do you know any book or paper dealing with plastic clamped thin membranes in the large deformation range? There must exist some engieneering analytic approach for this...

 

[Vigardo]

Besides Hencky´s formulas, I´ve found a recent (2008) paper from Zhao [3] with simpler equations for the stresses, strains and deflections of thin circular membranes. Although both approaches are very similar and their behavior is non-linear, they are not intended for plastic materials (only the Young´s modulus and the Poisson ratio are the linear material properties).
I´ve prepared and attached two plots with the comparison here:

Hencky_N: Hencky´s thin circular membrane adimensional stresses as a function of the adimensional radius (r/a) obtained by truncating the series in n = 0, 2, 4, 6 y 8 (purple, red, green, blue and black, respectively). Meridional and circunferential components are shown as full or broken lines, respectively. Zhao´s results are shown with yellow lines.

Hencky_W: Hencky´s and Zhao´s adimensional deflection as a function of the adimensional radius. The previous color scheme was used here again.

MORE REFERENCES:

[3] Zhao, Fuzhang. "Nonlinear solutions for circular membranes and thin plates", Proc. SPIE 6926, Modeling, Signal Processing, and Control for Smart Structures 2008, 69260W (April 03, 2008); doi:10.1117/12.775511; http://dx.doi.org/10.1117/12.775511
http://proceedings.spiedigitallibrary.org/proceeding.aspx?articleid=832283

 

[Chestermiller]

I have some ideas on what you can do to get a realistic model of this system, but I don't have time to discuss them right not. I'll get back with you later.

Do you have access to a biaxial testing machine like a Long Stretcher?

Chet

 

[Vigardo]

Ok, take your time. Thanks a lot for your help!

Sorry, I don´t have access to any testing machine.

Should I try FEA? Perhaps I should try to post in another forum?

In any case, the best for me would be having some kind of analytic equations instead.

 

[Chestermiller]

Ok, take your time. Thanks a lot for your help!

Sorry, I don´t have access to any testing machine.

Should I try FEA? Perhaps I should try to post in another forum?

In any case, the best for me would be having some kind of analytic equations instead.

This is a readily solvable problem. But the limiting factor is not how you solve the equations. The limiting factor is how accurately the stress deformation behavior of the real film can be parameterized in terms of the relationship between the two principal stress resultants and the two principal stretches. This relationship is going to be non-linear, considering the magnitude of the deformations that you are going to be dealing with. But, once this relationship is established, the rest of the problem is just math (although requiring numerical solution of a set of ODEs). You already have solutions for the case of small deformations of a linear elastic material. I know of no method of quantifying the required relationship based only on results from uni-axial stress strain measurements. Biaxial testing will be required for you to have confidence in your modeled results. However, if you are able to obtain biaxial test results on your material, I can show you how to get to your answer. Sorry about all this, but, in the real world of modeling physical systems, often the limiting constraint is being able to describe the material behavior.

Chet

 

[Vigardo]

Ok, so what I need is a stress-strain plot but for equal biaxial stress, i.e. not the one shown in post #1. Please, correct me if I´m wrong. Do you think that the film manufacturer would provide me this biaxial tests data?

Then let's suppose that I have valid biaxial testing data to describe Kapton HN behavior. Would you show me how to model the thin membrane?

Thanks a lot Chet!

 

[Chestermiller]

Ok, so what I need is a stress-strain plot but for equal biaxial stress, i.e. not the one shown in post #1. Please, correct me if I´m wrong. Do you think that the film manufacturer would provide me this biaxial tests data?

Then let's suppose that I have valid biaxial testing data to describe Kapton HN behavior. Would you show me how to model the thin membrane?

Thanks a lot Chet!

It's not just equal biaxial stress you need, although that would be much better than what you have. Don't forget that the circumferential stretch ratio at the holder is always equal to 1.0. So the deformations at different locations will run all the way from "uni-axial stretching with lateral constraint" to "equal biaxial stretching." Ideally, you need combinations of principal stretches such that 1≤λ_C≤λ_M. If you can obtain that data, you will be in good shape to do the large deformation modeling. I won't derive the model for you, but I will lead you through the steps in the derivation.

Chet

 

[Vigardo]

Great Chet! Thanks a lot!

So the data I need are some kind of maps (F and G) relating the strains (λ) with stresses (σ):
F(λ_C,λ_M)=σ_C
G(λ_C,λ_M)=σ_M
(Please correct me if I´m still wrong)

By the way, although I have some experience in MATLAB, I haven´t ever performed any numerical integration from ODEs. Perhaps I may need some additional help with this...

I would be very happy if you guided me in the derivation!

 

[Chestermiller]

Great Chet! Thanks a lot!

So the data I need are some kind of maps (F and G) relating the strains (λ) with stresses (σ):
F(λ_C,λ_M)=σ_C
G(λ_C,λ_M)=σ_M
(Please correct me if I´m still wrong)

Yes, but under the constraint on λ_M that I mentioned. Also, I would express F and G in terms of the engineering stress (force divided by initial cross section) rather than true stress. And I would carry out the stretching tests so that, in each test, the ratio of λ_C to λ_M is held constant during the sequence of increasing λ_M's.

Chet

 

[Chestermiller]

I want to modify what I said in my previous posting. In the stress tests, I'm recommending that λ_M and λ_C are parameterized as functions of time t as follows:

λ_M=1 + k t
λ_C=1 + k f t

where f is a constant between 0 and 1. f = 1 for equal biaxial stretching, and f = 0 for uniaxial stretching with lateral constraint. Most biaxial stretchers can be programmed to stretch in this way.

Chet