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Linear Viscoelasticity – Part 4 – Prony Spectrum

Introduction

A Prony Spectrum is a very useful concept that will help you calibrate and use linear viscoelastic models. Most people don’t know how simple it is to use a spectrum instead of all of those Prony terms. This article explains how to do it!

Prony Series

As mentioned in earlier parts of this series, a Prony series specifies how much the stress relaxes as a function of time in a stress relaxation experiment. Abaqus uses the following definition of the Prony series shear terms:

\[g_R(t) = 1 – \displaystyle\sum_{i=1}^N g_i \left( 1 – e^{-t/\tau_i} \right).\]

(1)

Ansys uses a similar (but not the same) expression:

\[g_R(t) = a_{\infty} + \displaystyle\sum_{i=1}^N a_i e^{-t/\tau_i}.\]

(2)

Other FE solvers use similar exponential-based expression. As I discussed earlier, similar expressions are also used for the volumetric response.

Here is an exemplar shear relaxation data set in Abaqus inp-file format:

				
					*Viscoelastic, time=prony
**       gi,     kappai,       taui
 0.00437895,          0,  0.0357113
  0.0831683,          0,    0.13542
   0.362453,          0,   0.513524
   0.362453,          0,    1.94733
  0.0831683,          0,    7.38442
 0.00437895,          0,    28.0023
				
			

The figure to the right shows that very little relaxation occurs at small and large times, most of the relaxation occurs for times close to 1 second. This is also reflected in the Prony series table.

Stress relaxation data

The problem with the Prony series that we need to find all of the\([t_i,g_i]\) values, and the \(g_i\) values are not independent. For example, \(g_4\) is likely to be between \(g_3\) and \(g_5\) in magnitude. There is a better way to do this: use a Prony Spectrum.

Prony Spectrum

If we plot the Prony series g-values as a function of time, the curve typically looks like Figure 1.

Figure 1. Analytical Prony series response specifying the g values as a function of time.

For use in a FE program the analytical spectrum response (shown in Figure 1) needs to be discretized into Prony points. Figure 2 shows one case with 7 Prony points, and one case with 10 Prony points.

Figure 2. Discretization of the Prony spectrum into a finite number of points.

Practical Use of a Prony Spectrum

MCalibration makes it really easy to use a Prony spectrum. The following equation is used to represent the spectrum (i.e. \(g(t)\) function):

\(g(t) = \displaystyle\exp\left[-\frac{1}{2}\left(\frac{t-t_m}{t_s}\right)^2 \right]^B,\)

(3)

where

  • \(t_m\) is the mean time for the spectrum
  • \(t_s\) is the standard deviation for the spectrum
  • \(B\) is a shape parameter.

The figure to the right shows how to select a Prony series spectrum in MCalibration.

Figure 3 (left) shows the stress relaxation response from a single Prony series term, and (right) the response from a Prony spectrum. The spectrum model has the same mean time, but a wider response due to the larger tstd value. The ability to directly set the mean and standard deviation of the relaxation response makes it very easy to use and calibrate a Prony spectrum model.

Figure 3. (a) Single Prony series response. (b) Prony spectrum response.

Summary

Using a linear viscoelastic Prony spectrum is in most cases better than specifying the Prony parameters directly.

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