Methods of interconversion between linear viscoelastic material functions. Part II—an approximate analytical method
Introduction
There are many times that one wants to predict one viscoelastic material function from another or to invert a Laplace or Fourier transform in a variety of problems. For example, it may be of interest to determine a transient material function, like relaxation modulus or creep compliance at very small times. It is often experimentally advantageous to get this from the corresponding complex material function in the frequency domain (obtained through a test with a steady-state sinusoidal input) rather than determining the transient function directly from a short-time relaxation or creep test. In some applications one may need the creep compliance when only the relaxation modulus is available or vice versa. Analysis of a viscoelastic continuum using the elastic-viscoelastic correspondence principle is based on the use of Laplace or Fourier transforms of related material functions to derive transformed response functions, which then requires transform inversion to predict time-dependent response.
In Part I (Park and Schapery, 1998) , we presented and tested a numerical method of interconversion between modulus and compliance functions when the given (source) and predicted (target) functions are based on a Prony (exponential ) series representation of transient functions. It was shown that the determination of a target function simply reduced to solving a system of linear algebraic equations for unknown Prony series coefficients, without the need to derive the target time constants. For ease of reference later in this paper, expressions in terms of Prony series constants for time and frequency dependent moduli are summarized in Appendix A.
In this paper we concentrate on approximate analytical interrelationships. It should be emphasized that, although the material characterization used here in the examples starts with a Prony series, such a representation is not needed. Indeed, it would be sufficient to start with numerical values of modulus or compliance obtained directly from an experiment ; some smoothing of the data may be needed because the slope in logarithmic coordinates is required.
Some existing methods are reviewed first. Then the narrow band property of the weight functions involved in various interrelationships between the material function is discussed. Based on these properties and the broad band representation of actual material functions, a set of new, easily applied, approximate analytical interconversions are developed and illustrated using a set of experimental data from polymethyl methacrylate (PMMA) . The new models are compared with existing models in their simplicity, accuracy, and limitations.
Section snippets
Some existing approximate interconversion methods
A large number of approximate, analytical interconversion methods with different bases and accuracies have been proposed by others (e.g., see Tschoegl, 1989) . Among these, a few methods will be selected and discussed, and then their performances compared with our new method.
Schapery (1962) presented two approximate methods of Laplace transform inversion, the direct method and the collocation method. As a special application of the direct method, the uniaxial relaxation modulus E (t) and the
Motivation and theoretical basis for the new method
Many of the approximate interconversion methods are based on different kinds of simplifications made in their original exact mathematical interrelationships. These simplifications are associated with the unique nature of the weight function involved in each interrelationship. From the theory of viscoelasticity, the following exact relations between two material functions may be obtained (e.g., Tschoegl, 1989) :
New approximate interconversion method
, , may be combined and rearranged to summarize the set of approximate interconversions :where the adjustment factors, λ̃, λ′, λ″, λ̂, λ̄ and λ∗ are given in Table 1 as functions of n. Relations (Eq. (35)) – (Eq. (40)) are exact when the material functions are
Numerical examples
We shall now illustrate the approximate interconversion method described above for PMMA material functions. The constants in the Prony series representation (A1) of the relaxation modulus of PMMA provide what we call exact representations of the material functions in (A1) – (A4) . These constants are the same as used in the companion paper (Park and Schapery, 1998) . Of course, it is not implied that these are exact representations of experimental data ; however, for a given set of constants,
Further discussion
Through the preceding examples for PMMA, we have seen that the new interconversion method improves results over the existing methods without adding any procedural complexity. However, in order to check the validity and accuracy of the method for uncommon cases of narrow-band material functions, two additional tests have been conducted.
As the first case, an artificial relaxation modulus E (t) simulating that of the foregoing PMMA, but using a 5-term (m = 5) Prony series representation (with
Conclusions
A new analytical method of approximate interconversion of linear viscoelastic material functions and approximate Laplace transformation and inversion ( for these and other functions) was introduced and its performance checked successfully using viscoelastic functions for PMMA. The method employs variable adjustment factors (or vertical shift factors on a logarithmic scale) dictated by the slope of the source function on a doubly-logarithmic scale.
Some available existing methods were reviewed
Unknown BIBs
Cost and Becker, 1970, Ferry, 1980, Schapery, 1961
Acknowledgements
Sponsorship of the contribution of the first author by the Office of Naval Research, Ship Structures and System, SandT Division, and the National Science Foundation through the Offshore Technology Research Center, is gratefully acknowledged.
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