Aka: Calibration and material modeling of the Payne Effect
This test data is digitized from the paper:
On the dynamical behavior of filled rubbers at different temperatures: Experimental characterization and constitutive modeling by DeLattre, et al. In this paper, a butadiene rubber filled with carbon black is considered.
The test data is attached in an Excel file. The paper only showed storage modulus, not the loss modulus or tan delta. The Excel file shows the data arranged in two different groupings, in order to make the two plots shown.
The Dynamic strain imported into the calibration app must be the Peak-to-Peak value.
So, task number one in this post is just to share some DMA test data that has both frequency sweep and a dynamic strain sweep information. Task number two is to discuss how one can use the Material Calibration app in 3DX to model these effects.
Can the Prony series capture the Payne effect? No, the Prony series linear viscosity cannot, by itself, capture the Payne effect. The Prony series model can capture the frequency dependence of the storage and loss modulus. But it cannot capture the dynamic strain dependence.
Can the PRF model capture the Payne effect? Maybe, to some degree. There is a fairly large body of scientific literature on modeling the Payne effect. Some authors talk about capturing the Payne effect with nonlinear viscoelasticity, and yes, the PRF model is a nonlinear viscoelastic model. But are the nonlinear viscoelastic equational forms in the PRF model the right ones to capture the Payne effect ? I would say the jury is still out on this question.
Calibration approach: Calibration of an advanced material model like the PRF model can be quite challenging. If possible, it may be helpful to approach the calibration incrementally. We do know that the Prony series can capture the frequency dependence of the storage and loss modulus, so an idea is to first perform that calibration to help identify some of the parameters, or at least "get in the ballpark". This image below is the calibration of the Prony series model, using R2021x FD05 (HotFix 5.13). This Prony work was done using the analytical mode since it is very fast. In analytical mode, the hyperelastic part must be the instantaneous modulus. The data was imported from Excel twice, using the different groupings to allow for creation of the two styles of plots. The SS-RO (Response Only) data and plot (upper right) was added to see the peak in the loss modulus versus frequency. This SS-RO data is at 0.001 dynamic strain.
The idea of first performing a Prony series calibration, then creating a degenerate (linearly viscous) PRF model from the Prony parameters, then doing a final calibration of the PRF model was discussed in good detail in this post: Calibration of a PRF Material Model for Polypropylene
After creating a 4-term Prony series model that generally captures the frequency dependence of the storage modulus, I map the Prony parameters to a set of degenerate 4-term PRF parameters. Since it is degenerate, I expect the degenerate PRF model response to match the Prony response. Here I have chosen to use the New Power Law creep model. It is helpful to turn plotting off as you enter the PRF parameters.
This is an image of the PRF model (below) after entering those degenerate parameters. We expected this degenerate PRF model to have the same responses as the Prony model and it does.
The exponent parameter, n, controls the degree of nonlinearity in the viscous behavior. As a first step, and largely to instruct, we set all of the n parameters to 2. (The RO data shown in purple has a dynamic strain of 0.001). Notice how the model begins to capture some amount of dynamic strain dependence.
With the above as the starting point, we run a calibration. The error norm is R2, the optimizer is Nelder-Mead. The DMA conversion parameters are the default values. The calibration result is shown in the image below. Notice that the calibration process has shifted the peak in the loss modulus to a significantly lower frequency.
Files in the zip file: these 3dxml files were created using R201x FD05, HotFix5.13 on April 4, 2021
DeLattre_paper_DMA_test_data.3dxml (just the data, imported twice for making nice plots)
DeLattre_paper_Prony.3dxml
DeLattre_paper_PRF_n=1.3dxml
DeLattre_paper_PRF_n=2.3dxml
DeLattre_paper_PRF_Final.3dxml
DMA_data_Fig6_Delattre_paper.xlsx
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