Duhamel Hereditary Integrals in Viscoelasticity

Mechanical behaviour of a viscoelastic model is described by its constitutive equation, i.e. the stress–strain relation. This constitutive relation is determined by the particular physical equations of both (elastic and viscous) elementary members of the model; and the geometric equations matching with the configuration of the model. Initially, the governing equation is of an implicit differential form of the order corresponding to the number of the irreducible viscous members involved in the model. Given the action as a function in time, we can directly derive the reaction on action dependence in an implicit differential form. Sometimes there is a need to have the reaction function explicit expression for a general action function. Constitutive equations enable us to study the mechanical response of the materials, represented by these models, to predict their mechanical response to a current load with regard to the load history and memory of materials.