Restrictions in a distributed complex fractional order linear constitutive equations of viscoelasticity

We analyze a viscoelastic body in a linear stress state by the use of the distributed complex order fractional derivative in the constitutive equation. A model is formulated so that it takes into account all complex order derivatives of the form t0Dtα+iβ(α), α∈[α1,α2]⊂(0,1), β∈C1([α1,α2]). Using a new procedure, we derive restrictions on a model that follows from the Clausius–Duhem inequality under isothermal conditions. We apply our analysis to the creep test, as well as to generalized wave equation on R, with a specific constitutive equation of Kelvin–Voigt type.