Some Equivalences in the Theory of Linear Viscoelasticity and Their Implications in Modeling and Simulation

Abstract

A class of linear viscoelastic models is examined which accurately represent the response of many materials in a wide frequency range. These include power law and logarithmic type models, such as those of fractional order, the Kuhn model of linear viscoelasticity and generalizations thereof. An equivalence between the fractional element and the generalized Kuhn model is established. A continuous representation of these models is naturally available. It is shown that a discrete rheological representation, in the form of series of springs and dashpots, is also possible. This allows an internal variable formulation, which establishes the thermodynamic admissibility of this class of models, in the sense that they satisfy the dissipation inequality. Introduction of appropriate state variables leads to convolution type equations, which, in the finite deformation case, retain much of the structure of linear viscoelasticity. Numerical implementation of the models is enhanced by the equivalence principles. It is shown that the accuracy expected of a long chain of classical structural units is achieved with a greatly reduced number of model parameters.