Cones of material response functions in 1D and anisotropic linear viscoelasticity

TL;DR

This paper characterizes the response functions of viscoelastic materials using cones, providing integral representations and mappings that relate scalar and tensor responses in 1D and anisotropic cases.

Contribution

It introduces a cone-based framework for describing viscoelastic response functions, extending the theory to both scalar and tensor-valued functions in 1D and anisotropic contexts.

Findings

Response functions satisfy cone conditions due to non-negative relaxation spectra

Integral representations fully characterize the cones of response functions

Mappings between response functions preserve cone structures

Abstract

Viscoelastic materials have non-negative relaxation spectra. This property implies that viscoelastic response functions satisfy certain necessary and sufficient conditions. It is shown that these conditions can be expressed in terms of each viscoelastic response function ranging over a cone. The elements of each cone are completely characterized by an integral representation. The 1:1 correspondences between the viscoelastic response functions are expressed in terms of cone-preserving mappings and their inverses. The theory covers scalar and tensor-valued viscoelastic response functions