Spectral Representation of Viscoelastic Materials

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. 

To introduce the spectral approach, consider the relaxation modulus for a generalized Maxwell model. 

The quantity E/AXi is similar to a Dirac delta function or singularity function (see appendix for a discussion) and is defined as H(X ). Taking the limit, Eq. 6.27 becomes. 

The quantity H(x) is a continuous function defined as the relaxation spectrum and is often used in polymer literature. (Note this term should not be confused with the Heavyside step function used earlier to represent a step input.) 

of course, the equation for a single Maxwell model and would only provide a very simple approximation of material behavior. The relaxation spectrum for a generalized Maxwell model for which 

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