On the Physical Meaning of Time-Domain Constitutive Models with Complex Parameters

TL;DR

This paper critically examines time-domain constitutive models with complex parameters, demonstrating they lack physical realizability and are only effective in frequency domain modeling of viscoelastic materials.

Contribution

It clarifies the physical limitations of complex-parameter models and shows that real-valued parameters are necessary for physically meaningful constitutive models.

Findings

Complex-parameter models are not physically realizable.

Relaxation modulus of complex-coefficient Maxwell model diverges at positive times.

Frequency-domain efficiency does not imply physical validity.

Abstract

This paper revisits the physical meaning of linear, time-domain constitutive models with complex parameters that have been presented in the literature and concludes that such models are not physically realizable. While complex-parameter phenomenological models (including those with complex-order time derivatives) may be efficient in capturing in the frequency domain the frequency-dependent behavior of viscoelastic materials over a finite frequency band, they do not possess physically acceptable time-response functions. The paper first reviews the intimate relation between the causality of a physically realizable constitutive model and the analyticity of its frequency-response function and explains that in theory it is sufficient to conduct a nonlinear regression analysis for estimating the model parameters either on only the real part or on only the imaginary part of its frequency-response function, given that they are related with the Hilbert transform. Consequently, the resulting model-parameters are real-valued; therefore, there is no theoretical justification to conduct the nonlinear regression analysis for estimating the model parameters in the complex space. The paper concludes with an example by showing that the relaxation modulus of the complex-coefficient Maxwell model is a divergent function at all positive times; therefore it is not a physically realizable constitutive model.