Linear viscoelasticity: Mechanics, analysis and approximation

Abstract: The aim of this review is to highlight the connection between well-established physical and mathematical principles as they pertain to the theory of linear elasticity. We begin by examining the physical foundations of Boltzmann and Volterra's hereditary law formalism, and how those principles restrict the form of the hereditary law. We then turn to questions of material stability and continuous dependence on the stress history within the framework of the Lax-Milgram theorem, which we find to set forth rigorous and unequivocal conditions for the well-posedness of the linear viscoelastic problem. The outcome of this analysis is remarkable in that it gives precise meaning to fundamental physical properties such as fading memory. Finally, we turn to the question of best representation of viscoelastic materials by finite-rank hereditary operators or, equivalently, by a finite set of history or internal variables. We note that the theory of Hilbert-Schmidt operators and $N$-widths supplies the answer to the question.