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Regularity lost: the fundamental limitations and constraint qualifications in the problems of elastoplasticity

Ivan Gudoshnikov

TL;DR

The paper develops a unified dual (constraint-based) framework for elastoplasticity, recasting quasistatic elasticity, elasticity-perfect plasticity, and elasticity-hardening plasticity as evolution problems governed by normal cones and sweeping processes. It demonstrates a sharp contrast between discrete models, which yield function-valued strains in both perfect plasticity and hardening, and continuum models, where a function-valued strain can fail to exist due to nonadditivity of normal cones and Slater-type constraint failures. By introducing constraint qualifications (Slater I/II, Rockafellar, Attouch–Brezis) and extended frameworks for hardening, the authors show that elasticity-hardening plasticity in $L^2$ can be well-posed with an additive decomposition of rates, under uniform linear growth of the elastic range. The work thereby clarifies when dual formulations produce solvable evolutions, connects discrete and continuum analyses, and points to future directions in limit analysis and higher-dimensional applications.

Abstract

We investigate the existence and non-existence of a function-valued strain solution in various models of elastoplasticity from the perspective of the constraint-based ``dual'' formulations. We describe abstract frameworks for linear elasticity, elasticity-perfect plasticity and elasticity-hardening plasticity in terms of adjoint linear operators and convert them to equivalent formulations in terms of differential inclusions (the sweeping process in particular). Within such frameworks we consider several manually solvable examples of discrete and continuous models. Despite their simplicity, the examples show how for discrete models with perfect plasticity it is possible to find the evolution of stress and strain (elongation), yet continuum models within the same framework may not possess a function-valued strain. Although some examples with such phenomenon are already known, we demonstrate that it may appear due to displacement loading. The central idea of the paper is to explain the loss of strain regularity in the dual formulation by the lack of additivity of the normal cones and the failure of Slater's constraint qualification. In contrast to perfect plasticity, models with hardening are known to be well-solvable for strains. We show that more advanced constraint qualifications can help to distinguish between those cases and, in the case of hardening, ensure the additivity of the normal cones, which means the existence of a function-valued strain rate.

Regularity lost: the fundamental limitations and constraint qualifications in the problems of elastoplasticity

TL;DR

The paper develops a unified dual (constraint-based) framework for elastoplasticity, recasting quasistatic elasticity, elasticity-perfect plasticity, and elasticity-hardening plasticity as evolution problems governed by normal cones and sweeping processes. It demonstrates a sharp contrast between discrete models, which yield function-valued strains in both perfect plasticity and hardening, and continuum models, where a function-valued strain can fail to exist due to nonadditivity of normal cones and Slater-type constraint failures. By introducing constraint qualifications (Slater I/II, Rockafellar, Attouch–Brezis) and extended frameworks for hardening, the authors show that elasticity-hardening plasticity in can be well-posed with an additive decomposition of rates, under uniform linear growth of the elastic range. The work thereby clarifies when dual formulations produce solvable evolutions, connects discrete and continuum analyses, and points to future directions in limit analysis and higher-dimensional applications.

Abstract

We investigate the existence and non-existence of a function-valued strain solution in various models of elastoplasticity from the perspective of the constraint-based ``dual'' formulations. We describe abstract frameworks for linear elasticity, elasticity-perfect plasticity and elasticity-hardening plasticity in terms of adjoint linear operators and convert them to equivalent formulations in terms of differential inclusions (the sweeping process in particular). Within such frameworks we consider several manually solvable examples of discrete and continuous models. Despite their simplicity, the examples show how for discrete models with perfect plasticity it is possible to find the evolution of stress and strain (elongation), yet continuum models within the same framework may not possess a function-valued strain. Although some examples with such phenomenon are already known, we demonstrate that it may appear due to displacement loading. The central idea of the paper is to explain the loss of strain regularity in the dual formulation by the lack of additivity of the normal cones and the failure of Slater's constraint qualification. In contrast to perfect plasticity, models with hardening are known to be well-solvable for strains. We show that more advanced constraint qualifications can help to distinguish between those cases and, in the case of hardening, ensure the additivity of the normal cones, which means the existence of a function-valued strain rate.

Paper Structure

This paper contains 43 sections, 29 theorems, 308 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries from convex analysis
  3. The abstract geometric framework and examples for linear elasticity
  4. An abstract linear operator and its adjoint
  5. The abstract geometric framework for linear elasticity
  6. Stress solution for elasticity in the abstract geometric framework
  7. Example 1 --- a discrete model with one-dimensional V
  8. Example 1 --- kinematics
  9. Example 1 --- Hooke's law
  10. Example 1 --- statics
  11. Example 1 --- the fundamental subspaces and the stress solution
  12. Example 2 --- a discrete model with two-dimensional V
  13. Example 2 --- kinematics
  14. Example 2 --- Hooke's law
  15. Example 2 --- statics
  16. ...and 28 more sections

Key Result

Proposition 2.1

Let $\mathcal{H}$ be a Hilbert space and $\mathcal{C} \subset \mathcal{H}$ be a closed, convex, nonempty set. For vectors $\bm{x}\in \mathcal{C}$ and $\bm{y}\in \mathcal{H}$ the following statements are equivalent:

Figures (12)

  • Figure 2: Schematic representation of the problem of Definition \ref{['def:ae']} and Remark \ref{['rem:abstract-operator-to-restrict']}. The unknown variables are indicated by blue color. In the problem of Definition \ref{['def:ae']} we are only looking for the unknowns $\bm{\widetilde{\varepsilon}}$ and $\bm{\widetilde{\sigma}}$.
  • Figure 3: Discrete models of Examples 1 ( a) and 2 ( b). Red arrows denote the external forces $\bm{F}$, applied at the nodes.
  • Figure 4: The fundamental spaces and the stress solution for elasticity in the discrete models of Example 1 ( a) and Example 2 ( b). The figure shows the situation with the stiffness parameters $k_i$ equal to $1$.
  • Figure 5: A continuum model of Example 3 in the relaxed reference configuration ( a) and an arbitrary current configuration ( b).
  • Figure 6: Schematic representation of the problem of Definition \ref{['def:aepp']}. The unknown variables are indicated by blue color. In the problem of Definition \ref{['def:aepp']} we are only looking for the unknowns $\varepsilon, \varepsilon_{\rm el}, \varepsilon_{\rm p}$ and $\sigma$. Red rectangles indicate the constitutive relations.
  • ...and 7 more figures

Theorems & Definitions (55)

  • Definition 2.1
  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.2
  • Theorem 2.1
  • Definition 3.1
  • Lemma 3.1
  • Remark 3.1
  • Definition 3.2
  • Remark 3.2
  • ...and 45 more