Prerequisite: Please read post "Understanding Prony series viscoelasticity - Part 1"
Converting from Time Domain to Frequency Domain Viscoelasticity
Converting from Storage and Loss Modulus measurements to Prony series viscoelasticity
Converting from DMA testing to Prony series viscoelasticity
Any of the above titles apply to this post. Remember that as soon as we mention Prony series, we are by definition limiting our discussion to linear viscoelasticity. In our Part 1 post we defined viscoelasticity as time-dependent elasticity, and we suggested several types of tests that can be performed, such as stress relaxation, or creep testing, to quantify this behavior. Another common test that is performed is called a DMA (Dynamic Mechanical Analysis) or DMTA (Dynamic Mechanical Thermal Analysis) test. There are many articles on the web describing these tests, such as wikipedia, or from companies that sell these test devices. In a nut shell, a DMA test excites a test specimen with a harmonic excitation and measures the in-phase and out-of-phase responses. The test may also sweep across a frequency range and/or a temperature range during the excitation. This test measures the energy stored in the material "Storage Modulus", and the energy dissipated by the material "Loss Modulus". These concepts are more fully explored in our classes on rubber: "Modeling Rubber and Viscoelasticity with Abaqus" and "Testing and Analysis of Elastomers" . Please attend one of these classes to get a broad understanding of elastomer and viscoelastic material modeling.
The purpose of this post is to focus on one particular issue in linear viscoelasticity that is often misunderstood and/or misused - the interconversion between frequency domain test data of storage and loss modulus and the Prony series. For instance, Abaqus allows one to input the frequency domain test data and it will automatically convert this information to a Prony series. Here is a snippet from the Abaqus Analysis Users' Guide (related to keyword usage *Viscoelastic, Time=Frequency Data) :
Frequency-dependent test data
The Prony series terms can also be calibrated using frequency-dependent test data. In this case Abaqus uses analytical expressions that relate the Prony series relaxation functions to the storage and loss moduli.
Conversely, one can specify *Viscoelastic, Frequency=Relaxation Test Data to indicate to Abaqus that a frequency domain analysis will be performed, but stress relaxation test data (time domain) will be used to define the viscous behavior of the material.
These interconversions are described in classic texts on viscoelasticity, such as Ferry's text Viscoelastic Properties of Polymers. However, one must remember that they are only valid for linear viscoelasticity, which is typically valid only at small strains, and only for certain types of materials.
While these interconversions can be handled automatically in the Abaqus software, we find that users often forget that the scope of the material testing must match the scope of the simulation. For instance, one might have some storage and loss modulus testing over a range of frequency from 1 to 500 Hz. It is possible to input that material test data and then perform a time transient analysis (*Visco or *Dynamic) to simulate what happens to a component as it creeps/relaxes over several hours, or even several days. It is possible to this, but bad practice, because the material was only tested over a very short time period - from your frequency based testing you have no idea how it will relax/creep over several hours, or several days. For this reason, we often recommend to our users that they only use frequency based material testing (DMA testing) for frequency based simulation, and they use time domain material testing (relaxation or creep) for time domain simulations. In this way it is much easier to keep track of the scope of the material testing versus the scope of the analysis.
This interconversion math is not all that complicated, so it is worthwhile to show what happens in an Excel spreadsheet. As I've said before, I like using Excel for this type of thing because it is so transparent (and ubiquitous). We can show test data, graphs of the test data and transparently show the equations used to convert the data. Please click on the links at the bottom to see a narrated video that describes this Excel spreadsheet with an interconversion from Prony terms to Storage and Loss moduli. The Excel spreadsheet is also attached for download. I'd like to thank my friends and colleagues at the SIMULIA Erie office for kindly providing this Excel spreadsheet.
There are some instructions on the first tab of the spreadsheet, it would be good to review these prior to viewing the video:
| Filename: Prony_to_Frequency_and_back_to_Prony.xlsx | |||||||||||
| Motivation: I wanted to test some of the calculations done in batch pre and to produce a "show and tell" spreadsheet that others could follow. This also makes clear the actual values that you need to input into A/CAE (or .inp file) to be converted back to Prony series. It also shows that you might get significantly different looking Prony terms, but the plot of the Modulus vs. time will be the same. | |||||||||||
| Instructions for logic: | |||||||||||
| 1) Start with some Prony g’s and tau’s (columns A&B on "Prony_to_Freq" tab, rows 3-12). | |||||||||||
| 2) Assume for this example that G_0 is 1.0 (hard coded into spreadsheet for now). | |||||||||||
| 3) Calculate storage & loss modulus values from Prony series using equations. | |||||||||||
| 4) Resulting Storage and Loss moduli are shown in columns H and I. | |||||||||||
| 5) Check plot of storage and loss modulus to see if frequency range is large enough (look for "flat" curves). | |||||||||||
| 6) Copy the green highlighted cells into the material definition in A/CAE. Look at image on far right. | |||||||||||
| 7) In A/CAE, run Evaluate, with Errtol =0.01 (default), copy results to the spreadsheet tab “Abaqus_1”. | |||||||||||
| 8) Run Evaluate with Errtol=0.001, copy results to the spreadsheet tab “Abaqus_2”. | |||||||||||
| 9) Plot the 3 responses on the "Prony_to_Freq" tab of the spreadsheet – they all match well. |
In order to manage the file sizes, we have broken this into two narrated videos. (The videos will play more clearly if downloaded first.)
Listen to segment 1 (7 minutes long) :
And segment 2 (9 minutes long):
Attached documents, for download:
Prony_to_Freq_and_back_to_Prony.xlsx (The spreadsheet shown in the videos).
Abaqus_CAE_files.zip (contains Abaqus_1.cae and Abaqus_2.cae files used for the Evaluate steps, steps 7 and 8)
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