Re-examination of the approximate methods for interconversion between frequency- and time-dependent material functions

doi:10.1016/j.jnnfm.2005.05.008

Journal of Non-Newtonian Fluid Mechanics

Volume 129, Issue 2, 30 August 2005, Pages 75-84

https://doi.org/10.1016/j.jnnfm.2005.05.008 Get rights and content

Abstract

The paper examines the correctness of the best-known approximate methods of interconversion between the frequency- and the time-dependent material functions. Approximate interconversions are compared to the close-form viscoelastic interrelations through the relaxation and/or retardation spectra. The analysis showed that the simple method, G(t) $≃$ G′(ω)|_ω=1/t, should be avoided. The most successful was the method of Schwarzl yielding results with maximum relative error within 2%. The magnitude of the error of interconversion is related to the magnitude of the response function first derivative. When the first derivative ψ = |d(log G′(ω))/d(log(ω))| < 0.5 all approximate methods except the simple method furnish interconversions within the acceptable error, i.e., less than 5%.

Introduction

Polymers are becoming increasingly important materials in mechanical-, electrical- and civil-engineering. In such applications, materials are expected to carry loads over extended period of time. This requires means of predicting their long-term reliability, which furthermore demands knowledge of viscoelastic material functions. To determine material functions in time-domain we need, in general, six independent experiments, three for force (or stress) excitation and three when the excitation is displacement (or strain). In addition, we need the same number of experiments to determine material functions in frequency-domain (dynamic material functions). These material functions are listed in Table 1.

Measurements in frequency-domain are common for characterization of melts, while the time-domain experiments are usually used for characterization of solids. Material functions in time and frequency-domain, within a given mode of excitation (displacement or force), are interrelated in a close form via the corresponding relaxation and retardation spectrum. The two spectra are not measurable directly; they must be calculated from the appropriate response function, measured either in time or frequency-domain. These calculations require solution of an inverse problem, which happens to be ill-posed. Until recently, there was no appropriate solution for these problems. Mainly for that reason several approximate methods for the interconversion between frequency- and time-dependent material functions have been developed in the past. Many of these algorithms are still in use, usually as part of the software packages supporting different apparatuses for the viscoelastic material characterization, e.g., rheometers and DTMA apparatuses. Most of these commercial apparatuses are dynamic, i.e., they measure material functions in the frequency-domain. The time-domain response is then predicted using one of the approximate algorithms [1].

The interrelations between frequency- and time-domain material functions are schematically presented in Fig. 1 [1].

The ultimate goal of this paper is to examine the correctness of the approximate methods of interconversion between the frequency- and the time-dependent material functions. We therefore compare the approximate interconversions to the close-form viscoelastic interrelations through the discrete relaxation and/or retardation spectra. The latter can be now readily calculated using one of the algorithms for calculation of mechanical spectra from the dynamic and/or static response functions, e.g., [2], [3], [4], [5], [6], [7], [8], [9], [10], [11].

We will first examine the approximate interrelations between frequency- and time-dependent material functions [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], using synthetic data generated from the known discrete spectrum. We carefully examine the effect of the shape of the spectrum on the accuracy of the interconversion. Next, we analyze the accuracy of approximate methods for four different materials, i.e., PVAc, NR, EPDM and PA66 [26], [27], [28].

Section snippets

Approximate interrelations

Mechanical spectrum cannot be measured directly. It has to be calculated from one of the response functions, usually from the harmonic (dynamic) response functions, e.g., G′(ω) and G″(ω). Dynamic measurements are most widely used in industry, because they are easily performed on one of the commercially available apparatuses.¹

Numerical experiments

We first examine the approximate interrelations between frequency- and time-dependent material functions using synthetic data generated from the known discrete spectrum H_ex(τ). Next, we analyze the accuracy of the approximate methods using measured relaxation functions, G(t), for four different materials. From them, we first calculate discrete spectrum H_ex(τ) using Emri–Tschoegl algorithm [6], [7], [8], [9], [10]. As in previous case, we consider that H_ex(τ) is the “correct” known spectrum.

Discussion and conclusions

The first set of analysis was performed on synthetic data generated from the four different synthetic spectra. In order to minimize the truncation error, the time span of spectra was chosen to be four decades broader than the corresponding response functions (see [6] for the detailed analysis of this problem). The spectra have been selected such that the corresponding response functions mimic the shape of the most commonly used polymers in solid and molten state, i.e., very slowly and very

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Re-examination of the approximate methods for interconversion between frequency- and time-dependent material functions

Abstract

Introduction

Section snippets

Approximate interrelations

Numerical experiments

Discussion and conclusions

J. Non-Newton. Fluid Mech.

Int. J. Solids Struct.

Int. J. Solids Struct.

Ill-posed problems in rheology

Rheol. Acta

A multidata method of approximate laplace transform inversion

Int. J. Numer. Methods Eng.

Determination of discrete relaxation and retardation spectra from dynamic moduli

Rheol. Acta

Generating line spectra from experimental responses. Part 1. Relaxation modulus and creep compliance

Rheol. Acta

Generating line spectra from experimental responses. Part 2. Storage and loss functions

Rheol. Acta

Generating line spectra from experimental responses. Part 3. Interconversion between relaxation and retardation behavior

Int. J. Polym. Mater.

Generating line spectra from experimental responses. Part 4. Application to experimental-data

Rheol. Acta

Generating line spectra from experimental responses. Part 5. Time-dependent viscosity

Rheol. Acta

Linearization as a method for determining parameters of relaxation spectra

Rheol. Acta

Theory of Elasticity