Numerical Approach to Prony Series Representation

A general issue in working with viscoelastic materials is representing the measured material properties by an appropriate mathematical function. As indicated earlier, a closed mathematical form facilitates solution of boundary value problems, as well as ease of manipulation of data. While viscoelastic properties can be represented by a number of functional forms, the exponential Prony series 

In the sign control method, as in several approaches, the first step is to select the relaxation times in a reasonable manner based on the time scale of the data. In such a process, the relaxation times are not chosen based on any known polymer structure or derived timescales, but are chosen for mathematical convenience. As real polymers contain a continuous distribution of relaxation times, in this approach a sufficient discrete subset of these relaxations are chosen in order to provide a mathematical function that will fit the material data. For a typical data set, such as that in Fig. 7.3, choosing the relaxation times evenly spaced in log time over the data range is reasonable. The number of relaxation times required varies depending on the smoothness of the data, but 10-20 relaxation times over 10 decades of time is a good rule of thumb. To facilitate fitting non-constant values at 

Once the relaxation times are selected, the problem reduces to finding the coefficients Ej such that the Prony series function optimally matches the provided time domain data. An obvious procedure to use is a generalized least squares approach, which was done in the multidata method (Cost and Becker, 1970). In this approach, coefficients are found that minimize the error between the modulus data (given as P data pairs (Ep,tp)) and the calculated function, E(t), 

Consequently, the sign control method (Bradshaw and Brinson, 1997), modifies the use of the least squares algorithm to ensure that the Prony coefficients be positive. This is accomplished via an iterative Levenberg-Marquadt method based on the first derivatives relative to each unknown coefficient (Press et al, 1992). The method is provided with an initial guess for the coefficients (all positive), uses these to predict a new set of values and then calculate If the new set decreases the error, it becomes the current step otherwise the previous values are used to take a smaller step. The additional constraint that Ej 0 is enforced by setting Ej - Ej before 

To illustrate the abihty of a generalized Maxwell Model (Prony Series) to fit long term data, consider the master curve data from Fig. 7.5 for polyisobutylene. A complete data set at 25°C was constructed as shown in Fig. 7.18. Thirty relaxation times evenly spaced in log time between 10 and 10 were chosen and the sign control method used to calculate the Prony series representation seen in Fig. 7.19. The modulus E(t) calculated from 

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