Fading memory (Q796323)

This paper is concerned with an excellent mathematical treatment of the concept of fading memory which plays an important part in describing the mechanical behaviour of a large class of simple materials. The original theory of fading memory, which has been introduced by \textit{B. D. Coleman} and \textit{W. Noll} [Rev. Modern Phys. 33, 239-249 (1961; Zbl 0103.408)], assumes that the domain of the tensor-valued constitutive functional on time is weighted \(L^ 2(0,\infty)\) space. However this large domain excludes some well established material models and observed phenomena. The present authors show that by considering smaller domains defined by different topologies (restricted class of deformation) hence larger topological duals containing distributions it becomes possible to incorporate rate-type theories into the theory of fading memory. Two cases corresponding to Sobolev spaces and to strictly denumerable inductive limits of Fréchet spaces are treated in detail, particularly for the linearized theories. The behavioural differences are illuminated by discussing the linearized shearing flow of an incompressible fluid.