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

Ivan Gudoshnikov

doi:10.3934/dcds.2026034

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2026. Doi: 10.3934/dcds.2026034

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

Ivan Gudoshnikov

Institute of Mathematics of the Czech Academy of Sciences, Prague, Czech Republic

Received: December 18, 2024

Revised: January 12, 2026

Early access: January 29, 2026

Published online: January 29, 2026

The author is supported by the Czech Science Foundation (GAČR) project GA24-10586S and the Czech Academy of Sciences (RVO: 67985840).

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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-plasticity with hardening 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 a 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 to stress constraints and the failure of constraint qualifications for them.

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.

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Mathematics Subject Classification: Primary: 74C05, 49J40; Secondary: 47J22, 47B02, 47B93.

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Figure 1. Examples of a sweeping process with moving sets $ {\mathcal{C}(t)\subset\mathbb{R}^2} $ having a) a smooth boundary b) a nonsmooth boundary

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Figure 2. Typical finite-dimensional situations where a pair of sets $ \mathcal{C}_1, \mathcal{C}_2 $ satisfies (see a) or does not satisfy (see b, c) Slater's constraint qualification (9). For the sets that appear in the elastoplasticity model (5)–(8), the corresponding condition (7) will be examined in Sections 4–5

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Figure 3. Examples of typical normal cones in a finite-dimensional setting. Here, we illustrate it with $ \mathcal{C} $ taken as a polygon in $ \mathcal{H} = \mathbb{R}^2 $. We depict the normal cones $ N_{\mathcal{C}}({\boldsymbol{x}}) $ as translated from $ 0 $ to $ {\boldsymbol{x}} $. Vectors $ {\boldsymbol{y}} $ are generic elements of $ N_{\mathcal{C}}({\boldsymbol{x}}) $, called supporting vectors to the set $ \mathcal{C} $ at $ {\boldsymbol{x}} $

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Figure 4. Schematic representation of the problem of Definition 3.4 and Remark 3.5. The unknown variables are indicated by blue color. In the problem of Definition 3.4, we are only looking for the unknowns $ {\boldsymbol{\widetilde{\varepsilon}}} $ and $ {\boldsymbol{\widetilde{\sigma}}} $

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Figure 5. Discrete models of Examples 1 (a) and 2 (b). Red arrows denote the external forces $ {\boldsymbol{F}} $ applied at the nodes

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Figure 6. 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 $

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Figure 7. A continuum model of Example 3 in the relaxed reference configuration (a) and an arbitrary current configuration (b)

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Figure 8. Schematic representation of the problem of Definition 4.1. The unknown variables are indicated by blue color. In the problem of Definition 4.1, we are only looking for the unknowns $ \varepsilon, \varepsilon_{\rm el}, \varepsilon_{\rm p} $, and $ \sigma $. Red rectangles indicate the constitutive relations

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Figure 9. The construction of the moving set in the discrete models of Examples 1.1 (a) and 1.2 (b). The figure shows the situation with the stiffness parameters $ k_i $ equal to $ 1 $

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Figure 10. With the particular data of Section 5.2, the moving set $ \mathcal{C}(t) \subset L^2_{{\bf C}^{-1}}(\Omega) $ consists of all functions constant a.e. on $ {\Omega = (-1, 1)} $ that fit in the gray area. a) illustrates the instant $ t = 1 $, after which the body force load and the shape of $ \mathcal{C}(t) $ will remain constant, and the evolution will be driven by the change of the displacement boundary condition. b) illustrates the instant $ t = t^* $ of the beginning of the plastic deformation. c) illustrates the instant $ t = 3 $ during the plastic deformation

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Figure 11. a) An illustration of the argument for Observation 1. b) An illustration of the argument for Observation 3

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Figure 12. a) As long as the blue rectangle is nondegenerate, we have that (154) and (153) are true, because any function $ {\boldsymbol{v}}\in L^\infty(\Omega) $ can be rescaled and translated to fit in the blue rectangle, cf (155). b) An explicit example showing that $ L^\infty(\Omega) $ is not closed in the $ L^2 $-norm. Take $ {\boldsymbol{y^*}}\in L^2(\Omega) $ as $ y^*(x) = (-x+1)^{-\frac{1}{4}} $, and approximate it by $ y_i(x) = \min(y^*(x), i), \, i \in \mathbb{N} $. We have $ {\|{\boldsymbol{y}^{*}}-{\boldsymbol{y}}_i\|_{L^2}(\Omega)\to 0} $, $ {{\boldsymbol{y_i}}\in L^\infty(\Omega)} $, yet $ {\boldsymbol{y^*}}\notin L^\infty(\Omega) $

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Figure 13. Schematic representation of the problem of Definition 6.7. The unknown variables are indicated by a blue color. In the problem of Definition 6.7, we are only looking for the unknowns $ \varepsilon, \varepsilon_{\rm el}, \varepsilon_{\rm p}, \sigma $, and $ \xi $. Red rectangles indicate the constitutive relations

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Figure 14. Elastic range (red interval), depending on the internal variable $ \xi $ for a particular $ x\in \Omega $. a) Elastic range expands nonlinearly in the plastic regime (isotropic hardening in particular). b) Linear growth condition (HA5a). c) Elastic range translates linearly, which is called linear kinematic hardening

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Table 1. The quantities in the examples of linear elasticity of this paper

Quantity	Example 1 (Fig. 5 a)	Example 2 (Figure 5 b)	Example 3 (Figure 7)
$ \mathcal{X} $	$ \mathbb{R}^3 $	$ \mathbb{R}^4 $	$ L^2(\Omega), \, \Omega=(a, b) $
$ \mathcal{H} $	$ \mathbb{R}^2 $	$ \mathbb{R}^3 $	$ L^2(\Omega) $
$ \mathcal{W} $	$ \mathbb{R}^3 $	$ \mathbb{R}^4 $	$ W^{1, 2}(\Omega) $
$ {\rm E}:\mathcal{W}\to \mathcal{H} $	$ \begin{pmatrix} -1 & 1 & 0 \\ 0 & -1 & 1 \end{pmatrix} $	$ \begin{pmatrix} -1 & 1 & 0 & 0 \\ 0 & -1 & 1 & 0 \\ 0 & 0 & -1 & 1 \end{pmatrix} $	$ {\boldsymbol{u}}\mapsto \frac{d}{d x}{\boldsymbol{u}} $
$ \mathcal{W}_0 $	$ \begin{array}{c} {\rm Ker}\, R, \\R={\begin{pmatrix} 1 & 0 & 0\\0 & 0 & 1 \end{pmatrix}} \end{array} $	$ \begin{array}{c} {\rm Ker}\, ({R} {\rm E}), \\{R} ={ \begin{pmatrix} 1 & 1 & 0\\0 & 1 & 1 \end{pmatrix}} \end{array} $	$ W_0^{1, 2}(\Omega) $
$ \mathcal{X}_0 =\overline{\mathcal{W}_0} $	$ {\rm Ker}\, R $	$ {\rm Ker}\, ({R} {\rm E}) $	$ L^2(\Omega) $
"Reactions space" $ \mathcal{X}_0^\perp $	$ {\rm Im}\, R^\top $	$ {\rm Im}\, (R{\rm E})^\top $	$ \{0\} $
$ {\bf E}: \mathcal{W}_0 \subset \mathcal{X} \to \mathcal{H} $	$ {\rm E} \text{ restricted to } \mathcal{W}_0 $	$ {\rm E} \text{ restricted to } \mathcal{W}_0 $	$ {\rm E} \text{ restricted to } \mathcal{W}_0 $
$ {\bf D}: D({\bf D})\subset \mathcal{H}\to \mathcal{X}_0^* $	$ \begin{array}{c} {\boldsymbol{\sigma}} \mapsto \text{the functional} \\ \left({\boldsymbol{u}}\in \mathcal{X}_0\mapsto \left({\rm E}^\top {\boldsymbol{\sigma}}\right)\cdot {\boldsymbol{u}}\right)\end{array} $	$ \begin{array}{c} {\boldsymbol{\sigma}} \mapsto \text{the functional} \\ \left({\boldsymbol{u}}\in \mathcal{X}_0\mapsto \left({\rm E}^\top {\boldsymbol{\sigma}}\right)\cdot {\boldsymbol{u}}\right)\end{array} $	$ {\boldsymbol{u}}\mapsto -\frac{d}{d x}{\boldsymbol{u}} $
$ D({\bf D}) $	$ \mathbb{R}^2 $	$ \mathbb{R}^3 $	$ W^{1, 2}(\Omega) $
$ {\bf C}:\mathcal{H}\to \mathcal{H} $	$ \begin{pmatrix} k_1 & 0\\ 0 & k_2 \end{pmatrix} $	$ \begin{pmatrix} k_1 & 0 & 0 \\ 0 & k_2 & 0 \\ 0 & 0 & k_2 \end{pmatrix} $	$ {\boldsymbol{C}}\in L^\infty(\Omega) $
$ \begin{array}{c}\mathcal{U}={\rm Im}\, {\bf E_C} \\ ={\rm Im}\, {\bf C E}\subset \mathcal{H}_{\bf C^{-1}}\end{array} $	$ {\rm Im}\, \begin{pmatrix} k_1\\-k_2 \end{pmatrix} $	$ {\rm Im}\, \begin{pmatrix} -k_1\\k_2\\-k_3 \end{pmatrix} $	$ \begin{array}{c}\left\{{\boldsymbol{\sigma}}\in L^2_{{\bf C}^{-1}}(\Omega):\vphantom{{\int\limits_\Omega}}\right.\\\left. {\int\limits_\Omega} \frac{1}{{\boldsymbol{C}}(x)}\, \sigma(x)\, dx=0\right\}\end{array} $
$ \begin{array}{c}\mathcal{V}={\rm Ker}\, {\bf D_C}\\={\rm Ker}\, {\bf D}\subset \mathcal{H}_{\bf C^{-1}}\end{array} $	$ {\rm Im}\, \begin{pmatrix} 1\\1\end{pmatrix} $	$ {\rm Im}\, \begin{pmatrix} 1 & 0\\1 & 1\\0 & 1\end{pmatrix} $	$ \begin{array}{c}\left\{{\boldsymbol{\sigma}} \in L^2_{{\bf C}^{-1}}(\Omega): \exists c\in \mathbb{R}\right. \\ \left. \text{for a.a. }x\in \Omega \quad \sigma(x)\equiv c \right\}\end{array} $
$ {\boldsymbol{g}}(t) $	$ \begin{pmatrix} 0 & l(t)\end{pmatrix} $	$ \frac{1}{2}\begin{pmatrix}1 & -1\\1 & 1\\-1 & 1\end{pmatrix} \begin{pmatrix} l_1(t)\\l_2(t)\end{pmatrix} $	$ \frac{\partial}{\partial x}{u_ {\rm D}}(t, x) $
$ \begin{array}{c}\text{Stress solution }\\{\boldsymbol{\widetilde{\sigma}}}(t)= G\, {{\boldsymbol{g}}}(t) + Q\, {{\boldsymbol{f}}}(t) \end{array} $	formula (54)	formula (71)	formula (90).

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Table 2. The quantities in the examples of elasticity-perfect plasticity, in addition to Table 1

Quantity	Example 1 (Fig. 5 a)	Example 2 (Fig. 5 b)	Example 3 (Fig. 7)
$ \Sigma\subset \mathcal{H} $	$ [\sigma^-_1, \sigma^+_1]\times[\sigma^-_2, \sigma^+_2] $	$ [\sigma^-_1, \sigma^+_1]\times[\sigma^-_2, \sigma^+_2]\times[\sigma^-_3, \sigma^+_3] $	$ \begin{array}{c}\left\{{\boldsymbol{\sigma}}\in L^2(\Omega): \text{ for a.a. }x\in \Omega \right. \\ \left.\sigma^-(x)\leqslant\sigma(x)\leqslant\sigma^+(x) \right\}\end{array} $
$ \begin{array}{c} \text{Moving set} \\ \mathcal{C}(t) \end{array} $	formula (128)	formula (132)	formula (136)

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Table 3. Constraint qualifications for different types of elastoplastic models

$ \textbf{Constraint qualification} $	$ \textbf{Discrete models} $	Continuous models in $ L^2(\Omega) $
		$ \textbf{perfect plasticity} $	hardening
		$ \textbf{perfect plasticity} $	nonuniform or sublinear growth of the elastic range, i.e. (HA5a) / (HA5b) fails	uniform linear growth of the elastic range i.e. (HA5a) / (HA5b) holds
Slater Ⅰ, Prop. 6.5 i)x	YES	NO	NO	NO
Slater Ⅱ, Prop. 6.5 ii)	YES	NO	NO	YES
Rockafellar, Prop. 6.5 iii)	YES	NO	NO	YES
Attouch–Brezis, Prop. iv)	YES	NO	NO	YES

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