Convert Prony Series to Frequency Data

Goal

The goal of this tutorial is to show how you can quickly convert the parameters of a  hyperelastic model with Prony series viscoelasticity into a table of strain amplitude, frequency, storage modulus (E’) and loss modulus (E”) data. The conversion will be performed using MCalibration.

Step 1: Specify the Material Model

Start MCalibration, and then open the material model dialog. Select the linear viscoelasticity model that you want to work with. In this example I will use the “Abaqus-Linear-Viscoelasticity” model. Specify the type of hyperelasticity and the number of Prony series terms. See image below.

Click OK to accept the material model. Back in the main window specify the known hyperelastic and Prony series parameters. The image below shows the values that I used.

Step 2: Specify the Frequency Domain Load Case

Create a text file that contains the following 5 columns:

  1. Mean strain
  2. Strain amplitude
  3. Frequency (either in Hz or rad/s)
  4. Storage modulus (E’)
  5. Loss modulus (E”)

Since we don’t have the storage and loss moduli, we can simply leave those values as zero. Here are the values that I selected for this example.

We are now ready to create the load case. Back in MCalibration, click on the + sign to open the load case dialog. Change the Load Case Type to “Dynamic Data (E’, E”)” See image below.

Click on the Load Experimental Data button and select the text file we created earlier. Then specify the columns so that they are in agreement with the text file we just created. See the following image. Click OK to accept the new load case.

Step 3: Update the Graph

In order to see the predicted dynamic data (storage or loss modulus as a function of frequency) we need to change the axes of the main graph. Click on the X or Y button above the graph to open the Axis Dialog box. Then in the dialog select the type of dynamic data that you would like to plot.

Step 4: Calculate the Dynamic Response

Close the change axis dialog. Then click Run Once in the main window. The results should then look like the following image. MCalibration has now calculated the dynamic properties at the requested strain amplitudes and frequencies.

Step 5: Save the Frequency Domain Results

The final step is to save the results. Click Save Predictions in the tool bar. Then click Select All, and then click Save. See the following image.

The exported results are save in a text file. Here is the file that I generated.

Summary

MCalibration can easily convert from a Prony series to dynamic properties at different frequencies. MCalibration can also easily convert from storage and loss moduli at different frequencies to a Prony series (with a hyperelastic model).

Fun Fact: The Prony series is named after the French scienties Gaspard de Prony, whos name is one of the 72 names that are engraved in the Eiffel tower. I will try to take a photo of that next time I visit Paris.

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