  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.
