Prerequisite: Please read the posts "Understanding Prony series viscoelasticity - Part 1 & Part 2"
This post builds upon earlier posts explaining details about the Prony series. From those earlier posts, and from the Abaqus course on rubber and viscoelasticity, calibration of a Prony series from time domain test data (stress relaxation test, or creep test) is explained in detail. The tool built into A/CAE for calibration of a Prony series from time domain test data generally works pretty well. There is also a tool built into A/CAE (and batch pre) to calibrate a Prony series from DMA test data - storage and loss modulus. However, that tool does not display graphical information, and in my experience, it often fails to find a valid set of Prony parameters (g's and tau's). In this post, we are limiting our scope to solid polymers, and as such, we believe that the material's bulk modulus, K, does not change with time.
Test Data
One of our biggest challenges at SIMULIA is obtaining test data that can be shown to others. For this post, we were fortunate to have access to this paper. It was also helpful that the paper showed test data as well as a fully developed Abaqus material model (in the appendix).
Juan-Sebastian ARRIETA, Julie DIANI, Pierre GILORMINI - Experimental characterization and thermoviscoelastic modeling of strain and stress recoveries of an amorphous polymer network -Mechanics of materials - Vol. 68, p.95-103 - 2014
This is an author-deposited version published in: http://sam.ensam.eu Handle ID: .http://hdl.handle.net/10985/7357
And the author kindly shared his test data with me (see figure 1 of the paper, or image below). I have modified the data slightly by removing some of the points. The DMA test data spans about 7 decades of time/frequency.
Errata: In creating this example problem, I have mistakenly called the test data Storage and Loss Shear moduli (G). In re-reading the paper, the DMA testing was performed using a uniaxial tension test specimen, and thus the data is Storage and Loss Tensile Moduli (E).
Excel Calibration Spreadsheet
I have set up a general purpose Excel spreadsheet for calibration of Prony coefficients from DMA test data. The spreadsheet is attached to this post, and there is a narrated video used to explain the contents of the spreadsheet. There is a worksheet used to store the test data, another worksheet is used to hold the Abaqus material model from the paper. Another worksheet holds the response from using A/CAE to calibrate the Prony series, and another worksheet shows that response in plotted form.
The worksheet that is used for the general purpose calibration is the first worksheet (leftmost) and is named "Initial Guess". It is named initial guess, because the approach used is to:
a) create an initial guess of the Prony terms of g_i and tau_i ;
b) use known equations to calculate the storage and loss modulus from the initial Prony guess, and
c) construct error measures between the test data and the calculated response.
d) Then an optimizer would be used to minimize the error measure(s).
I have used both the Excel optimizing solver and Isight to minimize the error norms. The 2nd video below explains the use of the Excel solver, and when it works, it is quite easy to use and fast. After running an error minimization process, I typically write out a new spreadsheet in which the coefficient values have been overwritten on the "Initial Guess" worksheet.
Excel Array Formulas
The formulas for calculating the storage and loss moduli from Prony coefficients became quite cumbersome, until a colleague of mine taught me how to use what are called Array Formulas in Excel. In the narrated video describing the Excel file we will point out the use of Array Formulas. The use of Array Formulas are facilitated by using named arrays for the g's (cell list named Gamma) and the tau's (cell list named Tau). Here is a good link to learn more about Excel Array Formulas:
http://office.microsoft.com/en-ca/excel-help/introducing-array-formulas-in-excel-HA001087290.aspx
Reformulation of the Basic Design Variables
In the initial version of this spreadsheet, the tau's were fixed (one tau per decade of test data) and the g's were the design variables. When cast in this way, one must also constrain the sum of the g's to be less than 1. In reviewing this spreadsheet, my colleague, Malik Kayupov, suggested a clever reformulation: create a new set of design variables, x_i, where each x_i lies between 0 and 1. Calculate the g's from these new design variables in such a way that the constraint of sum of g's < 1 is automatically satisfied. The power of this reformulation is that the minimization process converges much more quickly. One particular example is shown in the image below, the red line (Reform_NO) shows the minimization process over 20,000 iterations. The reformulated problem is shown in the black line (Reform_YES) and it converges to a slightly lower minimum in 1,521 iterations - over a 10x time savings!
Narrated video explaining the Excel spreadsheet: (This video is much clearer if downloaded first, 70Mb)
Narrated video showing the use of the Excel optimizing solver to minimize error norm, find Prony coefficients:
(also clearer if downloaded first, 49Mb)
Excel template spreadsheet for download:
Bonus video - setting up an Isight optimization of the Excel spreadsheet:
Back to: Material Modeling and Calibration - An Overview and Curriculum
