Identification of optimal history variables and corresponding hereditary laws in linear viscoelasticity

Abstract

We develop an operator-theoretic formulation of hereditary constitutive models and characterize optimal finite-rank internal-variable approximations in the sense of Kolmogorov $N$-widths. The history operator is shown to be compact under natural assumptions on the relaxation kernel, thereby admitting optimal low-rank approximations. The resulting reduced models inherit thermodynamic consistency, stability, and provable approximation bounds. An analysis clarifies the structural relation between hereditary representations and internal-variable theories and provides a rigorous basis for reduced-order modeling in computational mechanics. Selected numerical examples showcase optimal convergence of approximations with respect to rank and sampling.