Prony series and model selection of polymer materials

[lwn]

Dear All:

I am doing some simulation with ABAQUS for polymer materials. It is characterized by viscosity. In general, we use Prony series for viscous behavior. Can we say the \tao_1 as the "retardation time"? The second thing is if the strain is larger than 50%, can we still use Young's modulus (Hooke's law) for it? Or we MUST use hyperelastic model such as Mooney model. Right now, the Poisson ration is 0.45, the Mooney Rivlin model is for imcompressible material, can we use it? If yes, do you know how to convert the Young's modulus into C_1 and C_2 in Mooney model? In addition, I have only DMA test data of the polymer materials, how can I determine the suitable model for it? From the previous message, we'd better have uniaxial, biaxial test data or more data to select model, but now I could not get them, do you have any suggestion for model selection?

Thank you very much.

David

[sq]

David,

I hope you have a lot of luck with your project, but I must advise you that you won't be able to accurately model large-strain behavior using DMA information alone. DMA is typically done at strains of less than 1%. Unfortunately, the behavior of polymeric materials is very nonlinear at larger strains (50% elastic strain implies elastomeric behavior), and there is typically no way to predict accurately what will happen at larger strains based only on small-strain information.

You are right that the Mooney-Rivlin model would be appropriate for rubber behavior. A simpler model might be NeoHookean, and the Arruda/Boyce 8-chain model has a lot going for it. In any of these models (true of mathematical modeling in general), you need more data to calibrate the model to.

[lwn]

Hi, sg:

Thank you for your reply. My problem is I do not have other sample and data. What I have is DMA test data. I remember we can estimate the value of C_01 and C_10 of Mooney-Rivline model by means of Young's modulus, can we combine it with DMA test data and apply to large deformation analysis? For the relationship of C_01 and C_10 with E, I got G = 2(C_01 + C_10), but I do not find out other equation, it seems to be 4C_10 = E, do you have any idea for it?

Thank you.

David

[sq]

You could estimate one constant given only the Young's modulus, but certainly not two. If you assume C10=0, this effectively reduces the Mooney-Rivlin model to the Neohookean representation.

For uniaxial extension, where t is the force per unstrained area and L is the elongation ratio,

t = (L-1/L^2)*(C01+C10/L)

If I substitute strain into this, assuming that for small strain L = 1 + e, and take the derivative at e=0, then I get

E = dt/de|0 ~ 3(C01+C10)

I have no idea how you're going to get another equation.

[lwn]

Thank you, sg. Perhaps I should not say that is equation but a approximate relation, maybe 4C_10=C01 or other relation. I am not remember it clearly.

David

[Jorgen]

If you don't have more info about the material, I suggest simply using a compressible neo-hookean model.

- Jorgen

[lwn]

Hi, sg and Jor:

I am not familiar to polymer material. We know rubber imcompressibe. Can we consider most of polymer materials as imcompressible?

Thanks

David

[Jorgen]

:arrow: Rubber is (almost) incompressible at all strains
:arrow: Thermoplastics are compressible with a Poisson's ration about 0.3 at small strains, and almost incompressible at large strains (i.e. after yielding)
:arrow: polymer foams are compressible at almost all strains

[lwn]

Hi,

The second proximate relation is C01=4C10.

David

[lwn]

Hi, Jorgen:

Thank you. Right now I do not know the name of the polymer. Nooney-Rivlin is conditional stable. Do you know which model is non-condational stable, and how can we convert Mooney-Rivlin model parameters into that model parameters?

Thank you.

David

[sq]

I'll certainly defer to Jorgen, but this much is frankly very clear: you don't have enough data to support any model more complicated than neoHookean. Honestly, you don't really have enough to extrapolate the data you do have into the region you're interested in, even with this simple model.

Yes, there are other models that do not have the stability issues of M-R, but it doesn't matter. They are all the same for you: they are all unsuitable.

I'm really very sorry if this seems overly critical, but I'm trying to help you.

[lwn]

Hi, sg:

I know it. Should we have them, no problem. But if we don't, we have to try to find out one. For large deformation of polymer, we believe Hooke's law is not suitable and will lead to wrong results. Any other model like M-R, Neo-, Chain model will be better than Hooke's law. Results from them will be more approximate to real situation. It will be helpful before we can attain enough experimental data. From Young's modulus, we can obtain the approximate parameters of M-R model. Right now I do not mind the difference caused by the model. If I can convert M-R parameters into other model with unconditional stability, it will help me to obtain complete simulation at least. It is why I'd like to try to find out other model without enough data. Numerical simulation may go ahead before experiments.

Thank you.

David

[sq]

Well, David, it looks like you're certainly aware of the problem (I'm glad to see it). And yes, Hooke's law (linear elasticity) is probably not your best answer.

All this being said, the neoHookean model (not to be confused with Hooke's law) is still your best bet at this point. You don't really have independent M-R parameters, the "second equation" you quoted is really just a guess.

What's important to realize is that what the more complicated models do for you is correctly characterize the behavior at limiting stretch, where chain lengths between cross-links becomes an issue. This is explicitly true in the case of the Arruda-Boyce 8-chain model.

Unfortunately, unless the polymer is extremely heavily cross-linked, even moderate-strain data doesn't tell you much about this regime. Even at strains around 30%, the neo- model works adequately in most cases, and determining a second parameter is extremely difficult to do accurately. Is the limiting extension ratio 2? or 10? It's just not possible to tell.

[lwn]

Right now, we can predict 150% stain in polymer. It is much larger than 30% for NoeHookean model. It is hard to say NeoHookean model is enough.

David

[sq]

Well, David, good luck.

At least you know by now the pitfalls using a two (or more)-parameter hyperelasticity model with only low-strain data.