# MCalibration Prony Series

## Finite Element Definition of a Prony Series

In Abaqus the (shear) Prony series is defined by: $g_R(t) = 1 – \sum_{i=1}^N g_i \cdot \left( 1 – e^{-t/\tau_i} \right),$and in Ansys the (shear) Prony series is defined by:$g_R(t) = g_{\infty}+ \sum_{i=1}^N g_i e^{-t/\tau_i}.$Both of these equations are essentially the same,  and both have the following restrictions:

• The $$\tau_i$$ characteristic times have to be positive (non-zero) values.
• Each $$g_i$$ value has to be between 0 and 1.
• The sum of all $$g_i$$ values has to be between 0 and 1.

## MCalibration Definition of a Prony Series

The problem with the Prony series that is used by FE codes (and described above) is that there is a constraint not only on the individual $$g_i$$ terms but also on the sum of those terms. For many polymers the sum of the $$g_i$$ terms should be something like 0.9 to 0.99, and it is difficult for a non-linear optimization method to impose that constrains while at the same time searching for the individual $$g_i$$ terms.

Specifically, MCalibration is using the following relative $$g_i^{rel}$$ terms: $g_i \equiv \frac{g_i^{rel}}{g^{tot}},$ where $g^{tot} = \displaystyle\sum_{i=1}^N g_i^{rel}.$

So, if a Prony series has N terms, then MCalibration will use the following N+1 terms: $$g_1^{rel}, g_2^{rel}, …, g_N^{rel}, g^{tot}$$.

This variable substitution has the following important benifits:

• The $$g_i^{rel}$$ terms can have any positive value. They are not restricted to be less than 1.
• The $$g^{tot}$$ parameter has to be between 0 and 1.
• The material model calibration will typically be faster due to this parameter mapping.
• The parameters can be uniquely mapped back to the FE definition of a Prony series.