Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Linear viscoelasticity: Mechanics, analysis and approximation

Michael Ortiz

TL;DR

This work provides a rigorous, operator-theoretic foundation for linear viscoelasticity by unifying physical principles (causality, fading memory, reciprocity) with a variational, well-posed framework via the Lax-Milgram theorem. It shows that fading memory is essential for contractivity and stability of both local and global (boundary-value) problems, and it develops a comprehensive theory for representing and approximating memory via finite-rank (history/internal variable) operators, Hilbert-Schmidt properties, and $N$-widths. The analysis connects classical models (Maxwell-Wiechert, Prony series) with modern spectral and Hilbert-space techniques, providing convergence guarantees for kernel and spectrum approximations and a principled path to data-driven material identification. Overall, the paper furnishes rigorous criteria for stability, convergence, and optimal finite-rank representations, enabling robust model reduction and efficient memory-operator approximations in viscoelasticity.

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 viscoelasticity. 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.

Linear viscoelasticity: Mechanics, analysis and approximation

TL;DR

This work provides a rigorous, operator-theoretic foundation for linear viscoelasticity by unifying physical principles (causality, fading memory, reciprocity) with a variational, well-posed framework via the Lax-Milgram theorem. It shows that fading memory is essential for contractivity and stability of both local and global (boundary-value) problems, and it develops a comprehensive theory for representing and approximating memory via finite-rank (history/internal variable) operators, Hilbert-Schmidt properties, and -widths. The analysis connects classical models (Maxwell-Wiechert, Prony series) with modern spectral and Hilbert-space techniques, providing convergence guarantees for kernel and spectrum approximations and a principled path to data-driven material identification. Overall, the paper furnishes rigorous criteria for stability, convergence, and optimal finite-rank representations, enabling robust model reduction and efficient memory-operator approximations in viscoelasticity.

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 viscoelasticity. 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 -widths supplies the answer to the question.

Paper Structure

This paper contains 25 sections, 4 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Linear viscoelasticity
  3. Local state and evolutions
  4. The principle of superposition and hereditary laws
  5. Causality and time-shift invariance
  6. Instantaneous viscosity vs. delayed response
  7. The dissipation inequality
  8. Onsager's reciprocity principle
  9. The fundamental theorem of calculus and fading memory
  10. The local problem
  11. The local stress-control problem
  12. Existence and uniqueness of solutions
  13. Contractivity and material stability
  14. The viscoelastic boundary-value problem
  15. Approximation of the hereditary operator
  16. ...and 10 more sections

Key Result

Theorem 3.3

Assume: Then, problem (DxzJy5) has a unique solution $\epsilon \in \mathcal{M}$ and

Theorems & Definitions (17)

  • Example 2.1: Maxwell-Wiechert model
  • Definition 3.1: Local stress and strain spaces
  • Definition 3.2: Spaces of local stress and strain evolutions
  • Theorem 3.3: Local well-posedness
  • proof
  • Example 3.4: Maxwell-Wiechert model
  • Definition 4.1: Space of global evolutions
  • Theorem 4.2: Existence, uniqueness, continuous dependence
  • proof
  • Example 5.1: Approximation property of Prony series
  • ...and 7 more