The Frequency Response Function of the Creep Compliance

TL;DR

This paper develops a mathematical framework to accurately compute the frequency response function of creep compliance in viscoelastic materials, enabling better interpretation of rheometry data and model parameter identification.

Contribution

It introduces a method using generalized functions to construct the complex creep function, overcoming divergence issues and facilitating parameter extraction from experimental data.

Findings

Derived Hilbert pair relations for real and imaginary parts

Provided a technique for decomposing measured creep compliance

Enabled direct determination of viscoelastic model parameters

Abstract

Motivated from the need to convert time-dependent rheometry data into complex frequency response functions, this paper studies the frequency response function of the creep compliance that is coined the complex creep function. While for any physically realizable viscoelastic model the Fourier transform of the creep compliance diverges in the classical sense, the paper shows that the complex creep function, in spite of exhibiting strong singularities, it can be constructed with the calculus of generalized functions. The mathematical expressions of the real and imaginary parts of the Fourier transform of the creep compliance of simple rheological networks derived in this paper are shown to be Hilbert pairs; therefore, returning back in the time domain a causal creep compliance. The paper proceeds by showing how a measured creep compliance of a solid-like or a fluid-like viscoelastic material can be decomposed into elementary functions with parameters that can be identified from best fit of experimental data. The proposed technique allows for a direct determination of the parameters of the corresponding viscoelastic models and leads to dependable expressions of their complex-frequency response functions.