The image above comes from this post from Jan, 2014: Understanding Prony series viscoelasticity - Part 1
That post predates the Material Calibration App in the 3DExperience platform. It used an Excel spreadsheet to examine and explain the math behind the Prony series, in the context of stress relaxation. The plot above showed the behavior of just 1 term of a Prony series:
The initial shear modulus G0 is set = 1.0 The g_1 term (=0.2) represents the decay of the shear modulus over some time span. So, the shear modulus decays (decreases) by 20% over some time span. The time span over which the decay happens is given by the exponential function and for each Prony term is about +/- 1 decade in time and centered around the tau value. The user controls the overall span of decay by controlling the number of Prony terms. The post goes on to show that by adding together many Prony terms we can match many different shapes of stress relaxation behavior.
Related Posts:
Understanding Prony series viscoelasticity - Part 1 (Jan, 2014, Excel based post)
Understanding Prony series viscoelasticity - Part 2 (March, 2014, Excel based post)
Linear and Nonlinear Viscoelasticity (older version) (May, 2015, calibration performed with MCalibration)
Linear and Nonlinear Viscoelasticity (Feb 2021, calibration performed with the 3DExperience Material Calibration App)
The PRF model, which represents nonlinear viscoelasticity, was added to Abaqus in version 6.12, 2012. The Prony series represents linear viscoelasticity and has been part of Abaqus for decades. The mechanical analogy for a Prony series, or the PRF model, is the Generalized Maxwell Model. It consists of a single, purely elastic spring placed in parallel with multiple Maxwell elements (each consisting of a spring and a dashpot in series).
When I first began to calibrate the PRF model, I already had a very good understanding of the Prony model and an intuitive sense of how the G and Tau parameters changed the relaxation response. It was relatively easy to calibrate a Prony series.
In this post (link below), I used some mildly nonlinear stress relaxation test data and showed how to use the PRF model to capture the nonlinearity.
Linear and Nonlinear Viscoelasticity (older version) (May, 2015, calibration performed with MCalibration)
In that post, I introduced the idea of first calibrating a Prony series, then using the G and Tau values to populate a degenerate (linear) PRF model. I created an Excel file to make this conversion and various forms of it have been attached to other posts.
The PRF model's Network Stiffness Ratio (shown in the Excel file as SR1, SR2, etc.) plays the same role as the G term in the Prony series. It controls the amount of decay of the stress relaxation for that term.
For the classic Bailey-Norton Creep law, the PRF parameter "A" is related to the Prony parameters by A1 = 1/(6 * C10 * G1 * Tau1).
For this classic Bailey-Norton law, the N=1 and M=0 create the "degenerate" PRF model.
For this classic Bailey-Norton law, the N parameter controls the degree of nonlinearity as N increases, the creep strain rate also increases.
The image below shows a 1 term Prony response in the 3DExperience Material Calibration App. This essentially recreates the image at the top of this post. The red dots are some actual elastomer test data shown in that original post. This is not a calibration, it is just a response curve drawn for the hand-entered parameters.
Convert this to PRF:
Using the relationships from the Excel file, SRatio1 = G_1 = 0.2 ;
Using the new Power Law, Q0_1 = 1/A_1 = (6 * C10 * G1 * Tau1) = 200 MPa (units derive from C10)
Here is the PRF response :
I have used the "Material Calibration Case" feature in the Material Calibration App to show these response curves. Using this feature allows me to compare the Prony response to the PRF response. They are nearly identical.
One thing we have not touched on before is the effect of changing the PRF "m" parameter. For the Prony series, for a single term, one cannot change the shape of the curve relative to time. For the PRF one can by changing m. The image above has m=0.
In the three images below, we vary m from -0.1 to -0.3 to -0.5 In all cases the change in the Stress is 0.2 MPa. Changing the parameter "m" to larger negative values tends to spread the change over a larger time span.
The zip file below contains:
- The Excel file Prony_Example2.xlsx
- The Excel file Prony to Norton Conversion_3gs.xlsx
- A .3dxml file : Understanding_Prony_to_PRF_behavior.3dxml (from R2026x HotFix 4.29)
- A pdf file explaining the mapping of Prony to PRF parameters..
Part 2, Looking at the Stress Relaxation (SR) dataset
The above images showed a stress relaxation dataset for reference. In the image below we show a 8-term Prony material model that matches the stress relaxation over the range of time (1e-4 to 1e+4 secs). Tau_1 is 1e-4 and each tau after increases by 1 decade of time. Tau_8 is 1e+4. This fit is excellent with an R2 error norm = 0.99995
From the comments earlier in this post, and similar exercises in other posts, we know that we could construct an 8-term degenerate PRF model (n=1, m=0) and match this stress relaxation dataset. But can we get a good calibration with fewer PRF terms? The image below shows a 1-term PRF Power Law calibration in which we have kept the Neo-Hooke C10 value from the Prony calibration, left n=1 and the Reference Strain Rate =1. Notice the value of the m parameter. The R2 error norm is 0.995
Allowing the C10 parameter to be a design variable improves the R2 value to 0.9997
How about using the Bergstrom-Boyce law for the viscous network? We show below the equations for the Power Law versus the Bergstrom-Boyce Law.
Power Law Bergstrom-Boyce Law
Recall that in the Power Law we set a=0 (no pressure dependence).
In each law the leading parameter, the Reference Strain Rate, plays the same role.
In each law the parameter Reference Stress plays the same role.
The Power Law "n" parameter plays the same role as the B-B "m" parameter.
After spending several hours trying to calibrate a 1 term B-B model, I could not seem to "stretch out" the time frame of the decay.
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