Prony series

Prony series is the name given to a series of exponential terms, usually with the variable in the exponent negative and diminishing in absolute value. Reference is often made to R. Prony s paper in J. Ecole Polytechnique of 1795, but the derivation of the series that bears his name from his original paper is doubtful See note by Z. Rigbi, BulL Soc. Rheol., 23 (1), 1980,14... 

The relaxation curve of Fig. 18, although it refers to a model with 48 discrete relaxation times, can be very closely approximated by a Prony series collocated at Yd = 1,10 and 50 1020 to give... 

Another method of handing this would be by inversion of the Prony series, remembering that the product of the Laplace transforms of the modulus and the creep functions equals l7. 

The creep compliance function AD22(0 is modeled by using a Prony series of the form,... 

For systems in which the memory kernel decays fairly rapidly, it is not unreasonable to consider its approximation by a linear combination of exponential functions, a so-called Prony series ... 

We introduce a set of extended variables whose evolution is described by standard Brownian motions using Ornstein-Uhlenbeck (OU) processes. For this treatment, let the particle positions and momenta be enumerated as (qi,pi), i = 1,..., Ne and define the extended variable, associated with the kth term in the Prony series in interaction with the /th physical variable by... 

The relation between the Prony series representation and Langevin dynamics can be seen in an informal sense by multiplying the equation for Ui k equation by Xk, and considering Xk 0, giving... 

Comparison between linear viscoelasticity (one-term Prony series) and experimental data for GUR 1050 (30 kGy, y-Nj). 

Experimental data (-0.02/s, -0.05/s, -0.10/s) Linear Viscoelasticity (2-term Prony series)... 

S. W. Park and R. A. Schapery, Methods of Interconversion between Linear Viscoelastic material Functions. Part. I.-A Numerical Method Based on Prony Series Int. J. Solids and Struct. 36, 1653-1675 (1999). 

Due to the fact that non-Fickean diffusion is influenced by a change in the free volume of the adhesive, the viscoelastic bulk and shear compliances in the form of two separate Prony series are used. Poisson s ratio is allowed to vary with time for the adhesive. A shift factor definition based on the free-volume concept is used in the analysis. The adhesive is assumed to be initially moisture-free. A moisture concentration value of unity is specified on the adhesive boundary. 

This type of representation is sometimes called a Prony series and such an exponential expansion is often used to describe the relaxation modulus of a viscoelastic material even without reference to a mechanical model. 

In the solution of practical boundary value problems it is necessary to have knowledge of the actual creep or relaxation properties of the material. Sometimes experimental data in discrete form can be used in numerical solutions but most often measured values of E(t) or D(t) need to be represented mathematically. The most frequent mathematical approach to represent data is with exponential (Prony) series. The use of exponential series was well understood by early polymer scientist and polymer physicists who considered the need to mathematically represent data. However, as their focus was to develop understanding between macroscopic properties and molecular structure, they sought other general approaches that could be applied in a relatively simple fashion. While the resulting spectral approach may not appear simple, it has been widely used in polymer literature. 

If the time dependent modulus of the material, E(t), is expressed in a Prony series (generalized Maxwell model) representation (Eq. 6.31 or Eq. 5.22), then the simple algebraic form of the function leads to explicit expressions for the storage and loss moduli from solution to Eq. 6.49. 

Fig. 6.7(b) Associated relaxation spectra for poly isobutylene. Detail on the original data and the Prony series fit is contained in Chapter 7. 

An alternative approach to develop the TTSP expression (Eq. 7.25) is to consider an expression of the viscoelastic modulus as a Prony series as given by Eq. 5.21b or Eq. 6.25 with the temperature dependence now included on the basis of the theories of Rouse and Zimm. 

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... 

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),... 

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... 

Fig. 7.18 Master curve for tensile modulus of polyisobutylene at 25°C (Original data from Tolbolsky, (1972) and Catsiff and Tobolsky, (1955)). Fit from Prony series shown in Fig. 7.19.

Prony Series Coefficients for Catsiff and Tobolsky Data... 

Fig. 7.19 Prony series coefficients used to obtain tensile modulus for Polyisobutylene. Fit to data shown in Fig. 7.18.

Fig. 7.20 Experimental data from a DMA for polycarbonate shifted to form a master curve. Lines showing the fit of the Prony series from Fig. 7.21 are overlaid on the plot.

And with substitution of the Prony series in for E(t), one obtains the relaxation spectra as... 

Fig. 7.24 Relaxation spectra of polycarbonate calculated from the Prony series elements in Fig. 7.21 via Alfrey s rule.

As discussed in Chapter 7 real material properties extend over many decades of time and for realistic solutions of boundary value problems it is necessary to have methods to incorporate these real measured properties. When material properties can be represented by a Prony series composed of a number of terms, it is possible to obtain solutions for more practical representation of polymers. Examples of the use of Laplace transforms for... 

Discuss the power law in its various forms and compare it to the use of a Prony series to represent viscoelastic data. 

The master curve for a [90°]gs graphite/epoxy composite in uniaxial tension using TTSP is shown in Fig. 11.25. The following six-term Prony series representation of the data is also shown in Fig. 11.25 and as may be observed the agreement between the two is excellent. 

Fig. 11. 25 Comparison between a TTSP master curve and a six term Prony series for a [90°] graphite/epoxy composite at 160° C (320° F) and a stress of 35.7 MPa (5.18 ksi). Data from Hiel, (1984).

Fig. 11.26 Total stored and dissipated energy as calculated from Eqs. 11.58-11.60 normalized with respect to the initial total energy (t = 0 min) using the Prony series representation of the master curve in Fig. 11.25 Due to the log scale starting at t>0, the normalized total energy is slightly larger than one at the left end of the plot.

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