Rheological models vs. homogenization

Abstract

We show that the outcome of the theory of homogenization is at variance with a classical engineering technique, that is widely used for constructing rheological models of mechanical materials. Analogical Models. The constitutive behavior of several materials is often represented via analogical models, These models are discrete, and are built up by arranging few elements, that are representative of the basic behavior of the material, viz., elasticity, viscosity and plasticity. For univariate models (just in this case!), one may also use the pictorial image of parallel and series arrangements.These constructions are ubiquitous in engineering, see e.g. . In continuum mechanics they are usually named rheological models; analogous representations are used in electromagnetism, under the label of circuital models; in magnetism they are called magnetic circuits, and so on. As apparently these models are not derived from any representation of the mesoscopic structure of the material, here we address the following question: may the use of analogical models for continuous systems be justified by homogenizing an underlying lower-scale structure?(1)We anticipate that we shall answer in the negative, and will derive alternative models by means of the theory of homogenization. Although analogical models typically represent linear laws, here we are rather concerned with nonlinear relations: this will enhance the generality, without affecting the basic features of the question. Set-Up.In view of dealing with a periodic material, we might tile the Euclidean space R 3 by 3-dimensional cells [0, 1[ 3 . Equivalently, we shall assume the unit torus of R 3 , that we denote by Y, as a reference cell. As the period is typically very small, this normalization will accordingly need the use of a mesoscopic length-scale, when coupled with the macroscopic *