# Fit of initial_modulus vs log(strain_rate) master curve using prony series

Hi,

I am working on generating a strain rate based master curve from short-time tensile stress-strain data at different temperatures and strain rates.

After the initial fitting of tensile stress-strain data using 2 parameter equation, I got parameter initial youngs modulus vs strain rate for different temperatures. Afterwards, shifting along the horizontal axis (i.e, strain rate) was done to generate a Master curve for the initial youngs modulus. It was a success up till this point.

Moving ahead, I am required to fit this master curve (initial youngs modulus vs strain rate) using prony equation for the GMM model. Here is my fit for your kind reference:Â

Legends highlight 20 Maxwell elements for the GMM model each fitted considering each element per decade. In the end, 20 elements are fitted over the entire range of strain rate.

I did everything using Python. Ideally, the relaxation times for 20 maxwell elements should be in increasing order but this is not the case with my results.Â

If you see the graph below, which is relaxation times vs no. of maxwell elements, I experience certain waviness for elements ~11-16. I guess this is an abnormality and I did something wrong in my python algorithm written.

Could you please suggest something in this direction?

Thank you!

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Regards

Aman

I am not convinced that you did something wrong with your Python code. When I look at your master curve it seems that the curve seems to my eyes that they cannot be horizontally shifted to create a smooth continuous curve. That may be why you see he waviness. Another way to say it, there is now guarantee that you always create a smooth master curve. The results you see may be an indication that the material is not a linear viscoelastic thermo-rheologically simple material.

/Jorgen

Dr. Jorgen,

Thanks for your suggestions!

I missed sharing in my last post that the material I am calibrating is POM (polyoxymethylene) which is a thermo-rheologically complex material as you already pointed out. Because of this, I used both horizontal as well as vertical shifting. Is there some other technique that can be suggested for shifting in the case of thermo-rheologically complex materials?Â

Following is the master curve I used to fit prony series on. As you can see, little overlap is there in between shifted curves. Do you recommend me trimming these intersection points at higher strain rates hoping that it will help in generating better prony parameters? I used linear curves instead of original shifted curves.

Regards

Aman

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I would not trim the curve/points unless you have experimental reasons for it. I have seen many DMA results that look really bad at higher frequencies, so perhaps there is a reason for doing that here too, but I always try to avoid removing data that I don't like.

I would be interested in exploring a non-linear viscoelastic material here. That may fit your raw data better.

/Jorgen

Dr. Jorgen,

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Thanks for your suggestions again!

Now, I am using a non-linear viscoelastic model and I am getting the desired fit.

Afterward, I generated prony terms (Gi's and relaxation times) for shear modulus at different strain levels (considering relaxation phenomena). Used these as input in my UMAT. Using this UMAT, I carried out simple tensile tests in Abaqus for validation by comparing it with original experimental stress-strain curves. This is one of the results at a temperature of 80deg Celcius for your reference.Â

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I am working on a new material (an epoxy resin) now, which is also thermo-rheologically complex. The issue with this is that I have experimental data from DMA in terms of storage and loss modulus. For the generation of prony parameters from complex modulus, I followed the steps mentioned in one of your pdf guide and I am successful in implementing those steps using python and getting this fit for complex modulus vs frequency

Open point:

Since I would like to use the same UMAT as I used previously for POM material, I would like to generate prony parameters (Gi's and relaxation times) for shear modulus at different strain levels which I can directly input in my UMAT. Is there a way out which you can suggest to get relaxation shear modulus vs time plot at different strain levels?

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Best Regards

Aman

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