## 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.