Analyticity and causality of the three-parameter rheological models

In this paper, the basic frequency and time response functions of the three-parameter Poynting–Thomson solid and Jeffreys fluid are revisited. The two rheological models find application in several areas of rheology, structural mechanics, and geophysics. The relation between the analyticity of a frequency response function and the causality of the corresponding time response function is established by identifying all singularities at ω = 0 after applying a partial fraction expansion to the frequency response functions. The strong singularity at ω = 0 in the imaginary part of a frequency response function in association with the causality requirement imposes the addition of a Dirac delta function in the real part in order to make the frequency response function well defined in the complex plane. This external intervention, which was first discovered by PAM Dirac, has not received the attention it deserves in the literature of viscoelasticity and rheology. The addition of the Dirac delta function makes possible the application of time domain techniques that do not suffer from violating the premise of causality.