Rethinking the derivation of the constitutive equations of classical viscoelastic models using the Laplace transform method

Abstract Classical viscoelastic models constructed using linear springs and dashpots are the most common models for analyzing viscoelastic behaviors, and the core of these models is their constitutive equations. Compared to other derivation methods, the Laplace transform method is preferably chosen for deriving the constitutive equations of classical viscoelastic models because of the systematic nature of this method. In literature, the initial conditions associated with stress and strain are all assumed to be zero in deriving the constitutive equations of classical viscoelastic models using the Laplace transform method, but it is a paradox to make such an assumption, since the initial conditions are not necessarily all zero in general. In this article, we present the derivation of the constitutive equations of some classical viscoelastic models using the Laplace transform method under the assumption that the initial conditions are not necessarily all zero. It is believed that this is the most systematic and general approach for deriving the constitutive equations of classical viscoelastic models. The derivations demonstrate that the constitutive equation of a classical viscoelastic model is actually the same no matter the initial conditions are assumed to be all or not all zero. Hence, in practice, it is reasonable and recommended to assume the initial conditions are all zero for largely simplifying the derivation process when using the Laplace transform method for deriving the constitutive equation of a classical viscoelastic model.